1992 A, Aizpuru Advanced courses of mathematical analysis

A D V A N C E D C O U R S E S OF
MATHEMATICAL A N A L Y S I S
I
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EDITORS
A , AIZPURU-TOMAS
Fa L E O N - S A A V E D R A
Universidadde Cadiz, Spain
P R O C E E D I N G S OF T H E F I R S T I N T E R N A T I O N A L S G H O O L
A D V A N C E D C O U R S E S OF
MATHEMATICAL ANALYSIS I
Cadiz, Spain
N E W JERSEY
*
22-27 September 2002
LONDON
World Scientific
1
:
SINGAPORE
BElJlNG
*
SHANGHAI
HONG KONG
*
TAIPEI * CHENNAI
Published by
World Scientific Publishing Co. Re. Ltd.
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British Library Cataloguing-in-PublicationData
A catalogue record for this book is available from the British Library.
ADVANCED COURSES O F MATHEMATICAL ANALYSIS I
Proceedings of the First International School
Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or p u t s therevJ may not be reproduced in any form or by any means,
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V
Preface
An idea, a dream and then a reality. This is the manner we can describe
the gestation of the First International Course of Mathematical Analysis in
Andalucia.
Our small research group, working in the University of Cadiz, considers
that, nowadays, Andalucia has a remarkable scientific and cultural importance as well as an indisputable international presence and this idea should
take shape in some type of periodic event.
Our main aims were:
(i) That the different research groups in Andalucia working on mathematical analysis should meet each other and collaborate.
(ii) That the young researchers working in these groups could have access
to the most advanced research lines.
Besides, we consider it of great importance to unite efforts in order to
guarantee a solid education in our young researchers.
The Course was held in 2002, from September 23 to 27, in the historical
part of Cadiz, a beautiful city surrounded by the Atlantic Ocean.
This book is the first volume in a series of advanced courses of Mathematical Analysis. The authors of this collection are recognized experts with
an extensive research and educational experience. The authors of the first
volume are: Yoav Benyamini, Manuel Gonzblez Ortiz, Vladimir Miiller,
Simedn Reich (co-authored with E. Matouskova and A. J. Zaslavski) and
Angel Rodriguez Palacios.
The article by Benyamini is the only updated survey of the exciting and
active area of the classification of Banach spaces under uniformly continuous
maps.
The article by Gonzblez is a pioneer introduction to the theory of local
duality for Banach Spaces.
The paper, Genericity in nonexpansive mapping theory, by Eva Matoubkov6, Simedn Reich and Alexander J. Zaslavski, provides an up-to-date
detailed overview of the applications of the generic method to nonexpansive
mapping theory.
The article by Vladimir Muller provides a survey of results and ideas
concerning orbits of operators and related notions of weak and polynomial
orbits. The Scott Brown technique to obtain invariant subspaces is carefully
exposed.
The survey paper by Rodriguez Palacios collects the results on absolutevalued algebras since the pioneering works of Ostrwski, Mazur, Albert,
vi
and Wright to the more recent developments. The celebrated UrbanikWright paper on the topic is fully reviewed. The survey contains in addition
some new results, and several new proofs of known results. As a matter of
fact, it will be the authoritative reference for the optimal version of results
scattered in the literature through many separate papers.
We want to express our gratitude to all of them. We also want to thank
Professors Tomas Dominguez Benavides and Angel Rodriguez Palacios for
being the first in supporting the.idea of celebrating the Course.
We thank from the heart every participant for their presence: the mature researchers as well as the young ones still in their educational period.
Of all of them we keep pleasant memories.
Besides, we want to thank Professor Maria Victoria Velasco for assuming
the responsibility of organizing the second course in September 2004 in the
gorgeous city of Granada.
Finally, we want to thank the publishing house, World Scientific, for
making possible the interesting contents of our seminars to be enjoyed by
all the mathematical community.
The Editors
vii
Contents
Introduction to the Uniform Classification of Banach Spaces
Y. Benyamini
An Introduction to Local Duality for Banach Spaces
1
31
M. Gonzdez
Orbits of Operators
V. Muller
53
Genericity in Nonexpansive Mapping Theory
E. Matouikovci, S. Reach and A . J . Zaslavski
81
Absolute-Valued Algebras, and Absolute-Valuable Banach Spaces
A . R. Palacios
99
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1
INTRODUCTION TO THE UNIFORM CLASSIFICATION
OF BANACH SPACES
YOAV BENYAMINI*
Department of Mathematics, Technion
-
Israel Institute of Technology
Haifa 32000, Israel
e-mail: [email protected]. ac.il
This is an introductory survey of the classification of Banach spaces as metric
spaces, where the maps are (nonlinear) uniformly continuous maps or, more specifically, Lipschitz maps. We describe basic results which show that the uniform theory
and the linear theory are different but that, nevertheless, some linear features of a
Banach space are preserved under uniform homeomorphisms.
1. Introduction
The norm on a Banach space induces on it a topology and a metric. In
the linear theory of Banach spaces the natural maps are continuous linear
maps, and these maps “respect” not only the topology but also the metric
structure: they are bounded, i.e., Lipschitz maps. Of special interest are
the linear maps which preserve the metric, i.e., isometries.
One can, however, disregard the linear structure of the Banach spaces
and consider them as a special class of topological spaces or as a special
class of metric spaces. In this setup the maps are no longer required to
be linear and we consider continuous maps in the first case and uniformly
continuous, Lipschitz or isometric maps in the second.
The following two theorems define the “natural boundaries” for a meaningful theory. The first is due to Mazur and Ulam 32 and the second to
Kadec 26 (for separable Banach spaces) and to Toruliczyk 39 (for general
density characters).
Theorem 1.1. Let E and F be Banach spaces. Iff : E
isometry satisfying f(0) = 0, then f is linear.
-+F
is a surjective
‘This work was supported by the Technion Fund for the Promotion of Research.
2
Theorem 1.2. Two infinite-dimensional Banach spaces are homeomorphic
to each other iff they have the same density character.
These two theorems say that the topological and isometric classifications of Banach spaces are “trivial” in two opposite ways: Theorem 1.1 implies that the linear structure can be completely recovered from the metric
structure and thus reduces the (seemingly) nonlinear isometric classification problem to the linear one. Theorem 1.2 says that the topology gives
no information whatsoever on the linear structure.
The uniform classification of Banach spaces is lLinbetween”these two
extreme theories. As we shall see in these notes this is an interesting theory
which, on the one hand, gives some information on the linear structure of
the spaces but, on the other hand, it is rich enough to be genuinely different
from the linear theory.
A word of warning before we go on. While the topological or nonlinear isometric theories are “trivial” from the point of view of classification,
they are certainly far from being trivial. Infinite-dimensional topology is
a rich theory, see the books by Bessaga and Pekzy6ski lo and van Mill
40. Similarly, there are, of course, many problems concerning isometric (or,
more generally, 1-Lipschitz) maps, such as into-isometries, extensions, nonexpansive retractions, fixed points etc., which we do not consider here a t
all.
The purpose of this survey is to describe the main results, ideas and
techniques in the area. The idea is to draw the attention of the reader to
this exciting area and t o stimulate further reading and interest. We did
not try to give “best” results or to analyze variations of the concepts and
techniques. For these and for many more topics that are not discussed
here a t all we refer the reader to the recent book by the author and J.
Lindenstrauss. We also refer to for unexplained notation and terminology.
The article is divided to three sections. The first is a general description
of the area in the spirit of an expanded colloquium talk. In the other two
sections we describe in more detail and with proofs, or sketches of proofs,
some of the main results on the Lipschitz classification (in Section 3) and
on the uniform classification (in Section 4) of Banach spaces.
These notes are an expanded version of lectures delivered a t the “First
International Course of Mathematical Analysis in Andalucia” , which was
held in Cadiz, Spain, in September 2002. It is a pleasure to thank the
organizers and the participants of the course for the pleasant and inspiring
atmosphere. I also thank the referee for his useful comments.
3
2. A General Overview
Historically, the first result in infinite-dimensional topology is the following
theorem of Mazur 31. Denote the closed unit ball of L p ( p ) by B p ( p ) .
Theorem 2.1. Let p and u be two measures and let 1 5 p , T < 00. If L p ( p )
and L T ( u )have the same density character, then their unit balls, B p ( p )and
BT(u),
are uniformly homeomorphic t o each other.
The homeomorphism of B p ( p ) onto BT(p)(for the same measure p ) is
given explicitly by the map f -+ IflP/'sign(f). For different measures we
first use these maps pass to B2(p) and B2(u) and then note that these balls
are even isometric to each other.
Note that the L, spaces are not linearly isomorphic to each other for
different values of p . Hence the uniform classification (at least of the unit
balls of Banach spaces) is different from the linear classification of the
spaces. We shall see later that L, and L, (for p # q), as well as L, and 1,
(for p # 2) are also not uniformly homeomorphic to each other. Thus the
uniform classification of subsets of Banach spaces is also different from the
uniform classification of the whole spaces.
The next result, which generalizes Mazur's theorem, was proved for
spaces with an unconditional basis by Ode11 and Schlumprecht 35 and for
general lattices by Chaatit 12. For proofs and more details see also Sections
9.1-9.3 in '.
Theorem 2.2. Let E be a Banach lattice which does not contain 1,'s
uniformly. Then its unit ball is uniformly homeomorphic t o the unit ball of
a Halbert space.
The condition that E does not contain loo's uniformly is, in fact, also necessary. This was proved much earlier by Enflo 15.
Aharoni and Lindenstrauss gave the first example of two Lipschitz
equivalent Banach spaces which are not isomorphic to each other. It follows
that the Lipschitz classification of Banach spaces is different from their
linear classification.
We now describe the structure of the example. Let q : E -i F be a linear
quotient map from E onto F with kernel 2.A linear operator T : F -+ E
is called a lifting of q if qT = id^. Such a lifting is an isomorphism of F
into E whose image, T F , is complemented in E (by the projection Tq).
The kernel 2 is complemented in E (by the projection i d E - Tq) and E is
isomorphic to F @ 2.If T is a (nonlinear) Lipschitz lifting, then the same
4
formulas yield similar Lipschitz consequences: F is Lipschitz equivalent to
a Lipschitz retract of E , the kernel 2 is a Lipschitz retract of E and E is
Lipschitz equivalent to F @ 2.
The example of Aharoni and Lindenstrauss is a Banach space E which
does not contain a subspace isomorphic to @(I?) (for some uncountable set
I?), but such that there is a surjective linear quotient map q : E -+ @(I?)
with kernel co and such that q admits a Lipschitz lifting. It follows that
E is Lipschitz equivalent to %(I?) @ co = @(r),
but they are certainly not
isomorphic to each other: E does not even contain an isomorphic copy of
CO(F).
Godefroy and Kalton l7 have recently introduced a “categorical” construction, which, among other things, gave a systematic method of constructing more examples of the same nature. (See also Kalton 27 where
the approach is extended to spaces of Holder and more general uniformly
continuous functions.) We now describe the construction in 17.
Let E be a Banach space and denote by Lipo(E) the Banach space of
Lipschitz functions f : E -+ R satisfying f ( 0 ) = 0. The norm of f is its
Lipschitz constant. Let F ( E ) denote the closed linear subspace of the dual
Lipo(E)*,spanned by the point evaluations &(f) = f(~).The space F ( E )
is called the Lipschitz free space over E. Lipschitz maps between Banach
spaces induce, in a natural way, linear maps between their free spaces and
F ( E ) enjoys some useful properties of free objects in a category. The map
6~ : E + F ( E )is a (nonlinear!) isometry of E into F ( E ) . One checks easily
that the linear map PE : F ( E ) + E , given by P ~ ( z a , 6 , ~ )= Canz,, is
a surjective quotient map with llP~ll= 1. Clearly 6~ is an isometric lifting
of PE, hence F ( E ) is Lipschitz equivalent to @ ker(PE).
The next theorem yields the required examples. Recall that a Banach
space E is weakly compactly generated (WCG) if there is a weakly compact
subset K c E whose linear span is dense in E. Since the unit ball of a
reflexive space is weakly compact, reflexive spaces are certainly WCG.
Theorem 2.3. Let E be a nonseparable WCG space. Then F ( E ) does not
contain a subspace isomorphic to E . I n particular F ( E ) ,which is Lipschitz
equivalent to E @ ker(PE), is not isomorphic to it.
All the examples known to date of Lipschitz equivalent Banach spaces
which are not isomorphic to each other are nonseparable and nonreflexive.
The main open problem in this area is
5
Problem 2.1. If two separable Banach spaces are Lipschitz equivalent to
each other, are they necessarily isomorphic? What i f they are also reflexive?
Uniformly convex?
It turns out that the method used to construct nonseparable examples
cannot work in the separable case. This is one consequence of the following
remarkable theorem of Godefroy and Kalton 17.
Theorem 2.4. Let F be a separable Banach space and assume that it is the
image of some Banach space E under a surjective quotient map q : E --+ F .
If q admits a Lipschitz lifting, then it also admits a linear lifting.
In the linear theory the space Q is considered to be a “small” space.
For example, it does not contain reflexive subspaces or any of the other
classical Banach spaces. This is no longer true in the Lipschitz category.
Aharoni proved that it is actually “universal”. The proof of the following
theorem is presented in Section 3.
Theorem 2.5. E v e y separable metric space is Lipschitz equivalent t o a
subset of C O .
While we do not know whether the Lipschitz and linear classifications
of separable Banach spaces coincide, the uniform classification is certainly
different from both. The following result is due to Aharoni and Lindenstrauss 3 , improving on a previous result of R b e 38. For a proof, more
details and extensions see Section 10.4 in 8 .
Theorem 2.6. Let 1 I p , q , p n < 00 with p , --f p and let E = (C@lpn)q.
Then E is uniformly homeomorphic to E @ 1,. If p # q and p, # p f o r all
n, then E and E @ 1, are not isomorphic. If p = 1 and q , p , > 1 for all n,
then E is reflexive while E @ 11 is not, hence (see Corollary Z.l(i) below)
they are not even Lipschitz equivalent.
The results up to this point showed the difference between the uniform
and the linear classification. We now discuss some results in the opposite
direction, namely, instances in which at least some of the linear structure
is preserved.
The most natural way to “linearize” a mapping is by differentiation.
Recall that a mapping f : E --t F is said to be Giiteaux differentiable at
a point 2 if the limit Du = limt+o (f( x tu) - f ( x ) ) / t exists for every
u E E and is a bounded linear operator as a function of u. The operator
D is called the Gdteaux derivative of f at x and is denoted by Of(.).
+
6
If f is a Lipschitz mapping with constant K , which is G6teaux differentiable a t some point x , then the derivative D = D f ( x )is bounded by the
same constant K . Moreover, iff is a Lipschitz equivalence (Le., a Lipschitz
map which also satisfies the lower estimate 11 f ( x )- f (y)II 2 (12- y”/K for
every x , y E E ) , then D is also bounded from below by the same constant,
and is thus an into-isomorphism. (Note that D f ( x )is only bounded from
below. Its image may very well be a proper subspace of F even when f
was assumed to be a surjective Lipschitz equivalence.) It follows from this
discussion and from the next theorem that in many cases the existence of
a Lipschitz embedding of E into F implies that there is also a linear isomorphism of E into F . (The theorem, due independently to Aronszajn 5 ,
Christensen l 3 and Mankiewicz 30, will be discussed in detail in Section 3.)
Theorem 2.7. Let E be a separable Banach space and assume F has the
Radon-Nikodgm property (RNP). I f f : E t F is a Lipschitzfunction, then
there is a point x E E where f is G6teaux differentiable.
It is obvious that some assumption is needed in Theorem 2.7 on the
space F . For example, one cannot take F = co because by Theorem 2.5
every separable Banach space is Lipschitz embedable in CO, but “most”
Banach spaces are not linearly embedable in it. The assumption that F
has RNP is actually essential. Indeed, one of the characterizations of RNP
is that F has RNP iff every Lipschitz map from R to F is differentiable at
some point of R (or, equivalently, differentiable almost everywhere). Thus
for the theorem to hold we must assume that F has RNP.
Since reflexive spaces have RNP we obtain
Corollary 2.1. (i) If a separable Banach space E is Lipschitz equivalent to
a subset of a reflexive Banach space F , then E is isomorphic to a subspace
of F .
(ii) If E is Lipschitz equivalent to a subset of a Halbert space, then it is
isomorphic to a Hilbert space.
(iii) If p > 1 and r 2 1, then L, is Lipschitz equivalent to a subset of
L, iff it is isomorphic to a subspace of L,, i.e., iff r = 2 or 2 2 r 2 p .
Part (ii) was originally proved by Enflo
16. To deduce it from Theorem 2.7
we need the fact that E is isomorphic to a Hilbert space iff all its separable
subspaces are. Part (iii) also holds for p = 1, but this requires an additional
argument.
7
Differentiation results are, of course, not available for general uniformly
continuous maps. We now describe some of the techniques used in their
study. More details, including the definition and some basic facts on ultraproducts, will be given in Section 4.
A basic useful property of uniformly continuous mappings is that they
satisfy a Lipschitz condition for large distances. More precisely
Proposition 2.1. Let f : E -+ F be uniformly continuous. T h e n f o r every
a > 0 there is a constant K = K ( a ) > 0 such that [ l f ( z ) - f ( y ) ( [5 K[[z-y[(
whenever (111: - yl[ 2 a.
One way to apply the proposition is to use it to create Lipschitz maps
from uniformly continuous ones: Assume that f : E -+ F is uniformly
continuous and assume, as in the proposition, that I \ f ( z ) - f ( y ) I I 5 K J [ z - y l l
whenever 1 1 : - yJJ2 1. Put fn(z)= f(nz)/n.Then the map fn satisfies
Ilfn(z)- fn(y)II I K \ J z- y)) whenever JIz- y\I 2 l/n. It follows that if
U is a free ultrafilter on N,then g = (fn)uis a K-Lipschitz map from the
ultrapower ( E ) u into ( F ) u . Moreover, if f is a uniform homeomorphism,
then one can apply the same procedure to f-l. One checks directly that
( f n ) - l = ( f - l ) n , thus the procedure gives a Lipschitz inverse to g . We
have thus proved the following theorem of Heinrich and Mankiewicz 21
Theorem 2.8. If E and F are uniformly homeomorphic, then they have
Lipschitz equivalent ultrapowers.
As we shall see in Section 4, this result, together with the results of
Section 3, combine to give a simple proof of the following theorem of Ftibe 37.
Roughly speaking, the theorem says that uniformly homeomorphic Banach
spaces have “the same” finite-dimensional subspaces or, in the language of
Banach spaces theory, that they have the same local linear structure.
Theorem 2.9. Let E and F be two uniformly homeomorphic Banach
spaces. Then they are crudely finitely-representable in each other, i.e., there
8
is a constant C > 0 so that for every finite-dimensional subspace El of E
there is a finite-dimensional subspace Fl c F with d(E1,F l ) 5 C and vice
versa.
In particular, since L, and L, do not have the same local structure when
p # r (as follows, for example, by computing their type and cotype), it
follows that they are not uniformly homeomorphic to each other.
Proposition 2.1 can also be applied through the study of approximate
midpoints.
Definition 2.1. Fix x , y E E and 6
midpoints between x and y is
> 0. Then the set of &approximate
Mid(z,y,b)= { z : ) ) x - z l I , J ) ~ - z l l5
(~+~)IIxTYII/~}
When 6 = 0 we say that z is a midpoint (or, when we want to emphasize,
exact midpoint) between x and y.
It is clear that exact midpoints are mapped by isometries to exact midpoints. More generally, if f is K-Lipschitz and two points x and y happen
to satisfy l)f(x)- f (y))I = K J J x-yll, then f maps exact midpoints between
x and y t o exact midpoints between their images. The following proposition
generalizes this fact.
Proposition 2.2. Let f : E -+ F be a uniform homeomorphism and let
0 < 6 < 1/2. Then there are points z, y with 112 - yII arbitrarily large, so
that
f(Mid(GY76)) c Mid(f(x), f ( d , 5 6 ) .
Proof. We only sketch the proof and leave the exact computations, that
give the estimate 56, to the reader.
Let K ( a ) be the Lipschitz constant of f for distances larger than a.
Then K ( a ) is a nonincreasing function of a. Pu t K = lima-,mK(a).
Since f - l is also uniformly continuous, it follows that K > 0. Fix tu
so large that K ( a / 2 )
K . Choose x , y with llx - yII > a such that
(1 f (z) - f (y)(I K(lx- yII and let z E Mid(x, y, 6). Then
N
N
llf(4- f (.>I1
I K ( Q / 2 ) l I X - zll Kllx - 41
5 (1+ W l l x - Yll/2 (1 + 6)ll.f
(x)- f (Y)11/2.
9
We now explain the idea of how the approximate midpoint sets can be
used to show that some Banach spaces E and F cannot be uniformly homeomorphic to each other: Compute the approximate midpoint sets for points
in E and in F . By the proposition a uniform homeomorphism f : E -+ F
will have to take some approximate midpoint set in E into an approximate
fked point set in F . If it so happens that approximate midpoints sets in
E are “large” sets and approximate midpoint sets in F are %mall”, then
this would contradict the uniform continuity of f-’. To demonstrate how
this idea is implemented we shall compute in Section 4 the approximate
midpoint sets in 1, and L,, and then use the right notions of “large” and
“small”, appropriate for the different situations, to prove
Theorem 2.10. For every 1
5 p < 03, p # 2, the spaces L, and 1, are not
uniformly homeomorphic to each other.
This method was introduced by Enflo (unpublished), who used it to
prove the case p = 1 of the theorem. The case 1 < p < 2 is due to Bourgain
l1 and the case p > 2, which required a new notion of “large” and “small”
is due to Gorelik 19.
The problem of characterizing Banach spaces which are uniformly homeomorphic to a subset of a Hilbert space was completely solved by Aharoni,
Maurey and Mityagin 4. The result is, in fact, more general and holds
for linear metric spaces and not only for Banach spaces. For example, it
follows from the next theorem and from known results in the linear theory
that L p ( p ) is uniformly homeomorphic to a subset of a Hilbert space iff
05p52.
Theorem 2.11. A real linear metric space is uniformly homeomorphic to
a subset of a Hilbert space iff it is linearly isomorphic to a subspace of L o ( p )
for some measure p.
We shall not discuss this theorem in these notes. The interested reader
is referred to Chapter 8 in
The last problem that we discuss in this section is the uniform and
Lipschitz classification of balls and spheres in Banach spaces.
Benyamini and Sternfeld proved that for every infinite-dimensional
Banach space E there is a Lipschitz retraction from the unit ball B ( E )
onto the unit sphere S ( E ) . Equivalently, S ( E ) is Lipschitz contractible
and there is a Lipschitz map on B ( E ) with no approximate fixed point.
(The results in followed Nowak 34, who proved them for some special
10
spaces. See also Azagra and Cepedello-Boiso for a smooth version of
these results and for generalizations to starlike sets.)
This result leads naturally t o the following open problem.
Problem 2.2. Let E be an infinite-dimensional Banach space, are its unit
ball B ( E ) and unit sphere S ( E ) Lipschitz equivalent? Are they uniformly
equivalent?
The problem is open for any Banach space (including Hilbert space!) except
for one “pathological” counter-example: Gowers and Maurey 2o constructed
a separable reflexive Banach space, G M , which is not isomorphic to any of
its subspaces. For this space the unit ball cannot be Lipschitz equivalent
t o the unit sphere. Indeed, i f f : B -+ S were a Lipschitz equivalence, then
since the space is separable and reflexive f would be Giiteaux differentiable
at some point by Theorem 2.7. The derivative, D, would be an isomorphism
from G M into a proper subspace of itself (as is easy to deduce from the fact
that the image of f is in the sphere, which “looks” locally like a subspace
of codimension one). But there is no isomorphism of G M onto a proper
subspace of itself!
It is easy to see that for any Banach space E the unit sphere S ( E ) is
Lipschitz homogeneous, i.e., for any two points x,y in the sphere there is a
Lipschitz homeomorphism of the sphere onto itself taking x to y . It follows
that if the ball and the sphere of E are Lipschitz (or uniformly) equivalent,
then the ball should also be Lipschitz (or uniformly) homogeneous. The
only nontrivial advance on Problem 2.2 is the following result of Nahum 33.
See Section 9.4 in for a proof and further discussion.
’
Theorem 2.12. Let E be a Banach space which is isomorphic to E @ R,
then B ( E ) is Lipschitz (respectively, uniformly) equivalent to S ( E ) iff it is
Lipschitz (respectively, uniformly) homogeneous.
Here is a simple application of the theorem.
Corollary 2.2. Let E and F be two Banach spaces which are isomorphic to
E @ R and F @ R respectively. Then the balls B ( E ) and B ( F ) are Lipschitz
(respectively, uniformly) equivalent to each other iff the spheres S ( E ) and
S ( F ) are.
Proof. It is clear that if f is an equivalence between the spheres, then its
homogeneous extension gives an equivalence between the balls. Conversely,
assume that f : B ( E ) -+ B ( F ) is a Lipschitz (or uniform) equivalence. I f f
11
takes S ( E ) onto S ( F ) ,then we are done. So assume that there is a point
z 6 S ( E )such that Ilf(z)II < 1, and choose a point y E E with llyll < 1 and
llf(y)ll < 1. (Any point y with llyll < 1 which is close enough to z certainly
satisfies this condition.) Let g be a Lipschitz homeomorphism of B ( F ) onto
itself which takes f(z)onto f ( y ) . Then f-' o g o f maps the point z in
S ( E )to the point y E E with JJyJJ
< 1. Since pairs of points in S ( E ) can be
mapped to each other by a Lipschitz equivalence of B ( E ) and the same is
true for pairs in B ( E )\ S ( E ) ,it follows that B ( E ) is homogeneous and the
same is true for B ( F ) which is Lipschitz (or uniformly) homeomorphic to it.
By the theorem B ( E ) and B ( F ) are Lipschitz (or uniformly) homeomorphic
to S ( E )and S ( F ) repectively. Thus all the four sets involved are equivalent
to each other and, in particular, so are S ( E ) and S ( F ) .
0
3. Lipschitz Maps
The main result in this section will be the proof of Theorem 2.7 as well as
some further comments and variations. But we start with a proof, essentially due to Assouad 6 , of Theorem 2.5.
Proof of Theorem 2.5. A map f : X --t Q is given by a sequence of
real-valued functions f(z)= (fl(z),
fi(s),
...), where the fn's satisfy
(i) f,(z)-+ 0 for every z E X.
When X is a metric space, then f is a Lipschitz map iff
(ii) The fn's are Lipschitz with a common Lipschitz constant.
And f is a Lipschitz embedding if also
(iii) There is a constant C > 0 so that for every z # y in X there is a
n such that Ifn(2) - fn(Y)I 2 Cllz - YII.
+
In the construction we shall use functions of the form (a-d(z, M ) ) for
suitably chosen sets M C X and constants a. Such functions are Lipschitz
with constant 1.
To present the idea of the proof in a somewhat simpler setup, let us first
assume that X is compact and that its diameter is 1.
For each n 2 0 let {zy : i 5 m,} be a finite 2-,-net in X and put
f,,i(z)= (2-n+1 - d ( z , z y ) ) + . Then (i) and (ii) hold because Ifn,i(z)I5
2-n+1 + 0 and Ilfn,i((~ip = 1. To check (iii) fix z # y. Let n satisfy
2-(n+1) 5 d(z,y) 5 2-n and choose i such that d ( z , ~ 1 + <
~ )2-(n+3).
Then
fn+3,i(Z) = 2 -(n+3)+1 - d(z, ..+3)
- 2-(n+3)+l
>
- 2-(,+3) == 2-(n-k3) >
- d(z,y)/8.
12
On the other hand
2 d(z, y) - d(z, xy+3) > d(z, y) - 2-(n+3)
2 2-(n+l) - 2-("+3) = 3.2-(n+3)
d(y,
hence fn+3,i(y)
=0
and (iii) holds with C = 1/8.
For a general separable metric space X we need to replace the points
xy by sets. The main difficulty is that we cannot do this with finitely many
sets for each fixed n and this makes it more difficult to fulfill both conditions
(i) and (iii) simultaneously. We shall assume that X is a separable Banach
space. This is possible because every separable metric space is isometric to
a subset of the Banach space C(0,l).
Denote the unit ball in X by B. We construct a function F : X 4 co
with IIFl[m,
l l F l l ~ i5~ 1 so that FIB = 0 and so that 8 5 112 - yII 5 16
implies that llF(x) - F(y)II 2 112 - yll/l6. Once this is done put FO = F
and Fn(x)= 2nF(x/2n)for n # 0 in Z.We then write co as (Cz
@Q)O
and the required embedding is f(x) = (..., F-l(x),
Fo(x),Fl(x),...). To
see that the Fn's satisfy condition (i) notice that for each fixed x we have
F n(x) = 0 and also Fn(x)= 0 whenever n satisfies 2n > IIxlI. It
is clear that all the Fn's, hence also f, are 1-Lipschitz. To check condition
(iii) fix x # y and choose n such that 8 5 11(x - ~ ) / 2 ~ 5l l16. Then
IIFnk) - Fn(Y)II = 2"11F(x/2") - F(Y/2n)ll
2 2n[1(x- ~)/2~11/16
= llx - ~11/16.
To construct F let {Ai}i>obe a sequence of 1-balls that cover X . Denote
the concentric balls with radius 2 by 2Ai (and the ball of radius 2 centered
at the origin by 2B). Put MO= 2Ao \ 2B and
Mi
= 2Ai
\
( U M j ) \2B = 2Ai \ ( U 2 A . j )\2B
j<i
Then X \ 2B
(*)
=
for i 2 1.
j<i
u Mi and the disjoint sets Mi have the crucial property
For every x
E
X the set {i : d(x,M i ) < 1) is finite.
Indeed, fix k such that x E Ak. If d ( z , M i ) < 1, choose z E Mi with
I(x- zll < 1. Then z E 2Ak, hence, by the construction of the M's, cannot
belong to Mj for any j > k. Thus i 5 k.
+
Define now F = ( f o , f l , ...) : X -+ co by fi(x) = (1 - d ( x , M i ) ) .
Then Ilfillm, I I f i I l ~ iI
~ 1 and F is well defined because for every x E X
the condition (*) gives that f i ( x ) # 0 for finitely many i's only. Clearly
FIB EE0.
13
Finally, fix two points z , y with 8 5 1111: - yII 5 16. We may assume
that IIzII 2 2. If i is such that x E Mi, then fi(x) = 1. On the other hand
fi(y) = 0 because d ( y , M i ) 2 11z,yll - diam(Mi) 2 8 - 4 > 1. It follows
0
that fi(z)- fi(y) = 1 2 ( ( z- y11/16.
We now move to the discussion of Theorem 2.7. The proof of the theorem actually gives more than just one point of Giiteaux differentiability: the
function is Giiteaux differentiable “almost everywhere”. This requires an
explanation. There is no “good” measure on infinite-dimensional spaces (as
the next lemma shows), hence no natural notion of “almost everywhere”.
The heart of the proof of the theorem is the introduction of a family of
“negligible” sets in a general separable Banach space. These sets play the
role that sets of measure zero play in Rn and the theorem then says that the
function is Giiteaux differentiable on the complement of such a negligible
set.
Lemma 3.1. Let E be a n infinite-dimensional Banach space and let K be
a compact subset of E . T h e n there i s a point y E E so that all the translates
{ K t y : t E R} are pairwise disjoint. I n particular there is n o translation
invariant a-finite regular Borel measure p o n E .
+
Proof. The linear subspace V spanned by K is a proper subset of E.
Indeed, it is contained in the a-compact, hence proper (by Baire’s theorem)
subset UnL, where L is the closed convex symmetric hull of K . Choose
y E E \ V . If ( K t y ) n ( K sy) # 0, then there are XI,5 2 E K such that
z1 t y = 2 2
s y , i.e., y = (z1 - z2)/(s - t ) E V , a contradiction.
If p is a regular Borel measure, then there is a compact set K such that
p ( K ) # 0. Chose y as above for this set K . Being a-finite p cannot have
nonzero mass on all the uncountably many disjoint translates of K in the
direction of y. Thus p is not translation invariant.
0
+
+
+
+
Several different notions of negligible sets have been introduced. Aronszajn, Christensen and Mankiewicz introduced three such notions and another useful notion, the Gauss null sets, was introduced later by Phelps and
is, perhaps, the easiest to define. A Borel set A is Gauss null if p ( A ) = 0
for every non-degenerate Gaussian measure p on E . (A measure p is a
non-degenerate Gaussian measure if every functional 0 # x* E E x has a
non-degenerate Gaussian distribution with respect to p . ) The analysis of
the structure of negligible sets belongs to the area of infinite-dimensional
geometric measure theory and leads to many interesting results. (See Chapter 6 of for a systematic presentation of this topic.) It is a deep recent
14
result of Csornyei l4 that the two notions introduced by Aronszajn and by
Mankiewicz as well as the notion of Gauss null sets are actually equivalent
to each other.
We shall follow Christensen and will now present what he called “Haar
null” sets and their basic properties. This notion was introduced again
much later (under the different name of “shy sets”) in 22 (see also 23). We
work in separable Banach spaces, but the definition and the basic properties
of Haar null sets hold in any abelian Polish group.
Definition 3.1. A Borel subset A of a separable Banach space E is called
Haar null if there is a regular Borel probability measure p on E such that
p ( A z) = 0 for all z E E , equivalently, p * X A = 0. We denote the family
of Haar null sets by N.
+
Note that if E is finite-dimensional, then a set is Haar null iff its
Lebesgue measure is zero. Indeed, for sets of Lebesgue measure zero
just take p to be any probability measure equivalent to the Lebesgue
measure. Conversely if p is a test measure for A as in the definition,
then JJXA(Z
z ) d z d p ( z ) = J p ( A z ) d z = 0 , hence the inner integral
J x ~ ( z z ) d z is zero p-a.e. But this inner integral is identically equal to
the Lebesgue measure of A.
The family N is closed under finite unions. Indeed, if p i are test measures for {Ai}i5n,then p1 * ... * p n is a test measure for UAi. It is also
closed under countable unions. To prove this one needs to choose the test
measures pi so that the infinite convolution p = IIy * p i converges, and
then p is a test measure for UAi. See Proposition 6.3 in for details. It
follows that Haar null must have empty interior. Indeed, if A is open, then
E can be covered by a countable number of translates of A . Thus A E N
would imply that also E E N,which is false.
+
+
+
Proposition 3.1. If A is a Borel set and A $ N,then A - A contains a
neighborhood of 0.
Proof. Assume not and choose xj $ A - A with IIzjll < 2-j. Let C =
(0, l}N be the Cantor group (i.e., the group operation is addition modulo
2 in each coordinate). Put qn = (0, ...0, 1 , O ...), where the 1 is in the nth
position and denote the Haar measure on C by A. Define cp : C
E by
cp(d1,62, ..,) = C b j z j . The condition IIzjll < 2 - j implies that the series
converges and that cp is continuous. Since A $ N,it follows that there is a
point y E E such that X(cp-l(A y)) # 0. Indeed, otherwise the image of
--f
+
15
X under cp, namely, the measure defined by p ( B ) = X(cp-l(B)))could serve
as a test measure to show that A is Haar null.
A classical theorem in measure theory yields that the difference set
U = cp-l(A y) - cp-l(A y) contains a neighborhood of 0 in C, and thus
vn E U for large enough n. For such n the set cp-'(A+y) contains two points
that differ only in the nth coordinate, hence xn E (A+y) - (A+y) = A-A,
a contradiction.
0
+
+
Corollary 3.1. Let E be a n infinite-dimensional Banach space. Then
(i) If A is a Borel proper subspace of E , then A E N .
(ii) If A c E is compact, then A E N .
Part (i) is obvious, and (ii) follows from the fact that the compact set A - A
cannot contain an open set.
Since it is easy to construct, in any separable Banach space, a compact
set K and a Gaussian measure p such that p ( K ) # 0, it follows from (ii)
that the notions of Haar null and Gauss null are not equivalent.
An important property of Haar null sets is that they satisfy a weak
version of Fubini's theorem.
Lemma 3.2. Let V be a finite-dimensional subspace of E and denote the
Lebesgue measure on V by A. W e also use the same notation X for its
translates to translates of V . I f X(A n {V y}) = 0 for all y E E , then
+
AEN.
Proof. As a test measure just take any probability measure on V which is
equivalent to A.
0
To prove Theorem 2.7 we shall show that there is a Haar null set A such
that f is GSteaux differentiable on E \ A. As a preparation we first present
a simple technical lemma and then a slight generalization of the classical
theorem of Rademacher that a Lipschitz map between finite-dimensional
spaces is differentiable almost everywhere.
Lemma 3.3. Let G be a dense additive subgroup of E and let x E E . If
f : E -+ R is a Lipschitz function such that limt+o (f( x tu) - f ( x ) ) / t
exists for all u E G and is a n additive function of u E G, then f is Giteaux
differentiable at x . I n particular the set of Gdteaux differentiability points
o f f is a Borel set.
+
16
+
Proof. The functions ht(u)= (f(z tu) - f ( z ) ) / tare Lipschitz with the
same constant as f. Hence their convergence on the dense set G implies
that the limit exists everywhere. The additivity of the limit is clear, and
the homogeneity follows from a change of variable: ht(au)= ah,t(u).
To see that the set of differentiability points is Borel we choose the dense
group to be countable and fix any u E G. P u t ht(z,u)= (f( z + t u ) - f ( z ) )/ t
and denote the set { ( s , t ) E Q x Q : Isl,Itl < l/m} by Dm. Then the
derivative a t z o E E in the direction u exists iff
zo E
nun
{z :
-
<1
~ ~ 1
n m D,
and this set is Borel. Since G is countable, the intersection of these sets
over all u E G is also Borel. The additivity in u E G is also given by a
countable set of restrictions.
0
Proposition 3.2. Let F be a Banach space with RNP and let f : R"
be a Lipschitz function. Then f is differentiable almost everywhere.
-+ F
Proof. Let cp : R" -+R be a non-negative smooth function with compact
support such that J cp = 1. Let G be a countable dense additive subgroup
of R" and for each u E G put $,(z) = limt+o (f(z tu) - f ( z ) ) / t .As F
has the RNP, f is differentiable in the direction of u almost everywhere on
every line parallel to u,hence almost everywhere in R". Moreover, since G
is countable $,(z) exists for almost all z simultaneously for all u E G. It
remains t o show that it is additive as a function of u E G.
P u t g = f * cp, the convolution of f and 9. Then g is smooth and its
derivative a t z applied to u,namely D,(z)u = f * D,(z)u, is linear in u.
On the other hand
+
+
D,(z)u = lim (g(z tu) - g ( z ) ) / t
t-0
(The passage from the first line to the second follows by Lebesgue's dominated convergence theorem.) Combining this formula with the linearity of
D g ( z )we obtain that cp * ($,+, - $, - $,,) = 0 for all u,v E G. Replacing
cp by cpj(z) = j"cp(jz) and letting j
00 gives that $
+
,,
- ($,
$,,) = 0
a.e. for all u , v E G.
0
--f
+
Proof of Theorem 2.7: Let Vl c Vz c ... be finite-dimensional subspaces
of E whose union is dense in E and let D, be the set of all 3: E E such that
17
+
the limit limt,o (f(x t y ) - f (x))/ t exists for every y E V, and is linear
as a function of y E V,.
By Proposition 3.2 D, n {V, y} is a set of full measure in V, y for
all y E E , i.e., X((E \ D,) n {V, y}) = 0 for all y E E . By lemma 3.2 the
set E \ D, is Haar null for every n, and since N is closed under countable
unions U ( E \ D,) E N . On no,, the complement of this negligible set, f
is Ggteaux differentiable.
0
+
+
+
We shall also need an analogous theorem on w*-derivatives. Let E
and F be Banach spaces and let f : E --+ F* be a Lipschitz map.
We say that f is w*-differentiable at a point x E E if the w*-limit
Dj(z)y = w*- lim (f(x t y ) - f (x))/ t exists for every y E E and if Dj(z)
is a bounded linear operator from E to F*. Since the norm is w*-lower
semi-continuous, it follows that the norm of D ; ( z ) is bounded by the Lipschitz constant of f. Unfortunately, even when f is a Lipschitz equivalence
Dj(z) does not have to be bounded from below, i.e., an into-isomorphism.
Nevertheless, Heinrich and Mankiewicz 21 proved that it is bounded below
almost everywhere, i.e., on the complement of a Haar null set.
+
Theorem 3.1. Let E be a separable Banach space and let f : E + F* be
a Lipschitz embedding of E into a dual space. Then f is w*-differentiable
almost eve rywhere and its derivative is bounded from below almost everywhere. More precisely, i f llx - yII
11 f).( - f(y)(I Kl(z - y ( (for every
x,y E E and i f b < 1, then for almost every x E E the w*-derivative o f f
exists and satisfies llD;(x)uII 1 bllull for every u E E .
<
<
Proof. The proof of the almost everywhere w*-differentiability is similar
to the proof of Theorem 2.7 and we omit it.
Instead of proving the almost everywhere lower boundedness we shall
only indicate why it is true by describing an analogous but simpler setup.
Assume that f : R + R satisfies 1s - tl
I f ( s ) - f (t)l 5 Kls - tl for
every s , t E R and that b < 1. We show that f’, which is known to exist
almost everywhere, satisfies If’(t)l 2 b almost everywhere. Indeed, put
A = { t : If’(t)l < 6). If X(A) > 0, then A has a density point and we can
chose, for any E > 0, an interval I = [a,b] such that X(An1) > ( l - c ) ( b - a ) .
Since If‘l is bounded by K we obtain
<
1s
b
b -a 5
If@)
-
f ().I
=
f’(t)l
18
provided E is small enough. A contradiction.
The estimates in the proof of the theorem are completely analogous, see
Theorem 7.9 in for details.
0
As remarked in the introduction, the derivative (or the w*-derivative)
is only bounded from below. Thus even if f is assumed to be a surjective
Lipschitz equivalence it does not follow that the derivative, at a point where
it exists, is necessarily surjective. (A concrete example of such phenomenon
was given by Ives and Preiss in 24.) The following problem is open
Problem 3.1. Assume that f : E + F is a surjective Lipschitz equivalence
and that E and F are separable. Is there a point xo where D l ( x 0 ) is a
surjective isomorphism? This is unknown even when we add assumptions
on E and F such as that they are reflexive or uniformly convex or even
Hilbert spaces.
There are two natural approaches to this problem, but they both fail:
(i) A positive answer would follow if f had to be F’rkchet differentiable
at some point. But this is false. If ?I, : R + R is any Lipschitz equivalence with $(O) = 0 which is not differentiable at 0, then f(a1,a2, ...) =
($(al),$(a2), ...) is a Lipschitz equivalence of 12 onto itself which is not
F’r6chet differentiable anywhere.
(ii) If A E N would imply that f(A) E N,then we could apply
Theorem 2.7 to both f and f-’ and find a point xo such that f is
Ggteaux differentiable at 20 and f - l is Gdteaux differentiable at f (20).
It would then follow by the chain rule that the derivatives at these points
are inverse to each other. But Lindenstrauss, MatouSkov6 and Preiss 29
(see also Theorem 6.14 in ) showed that a Lipschitz equivalence does not
have to take N into itself. (Their article contains, in fact, stronger results
that yield similar consequences for the other families of negligible sets and
not only for the family of Haar null sets.)
It thus seems that differentiation is not sufficient by itself to solve Problem 2.1. We shall now combine it with information on projections, which
will allow us to show that in many cases if two Banach spaces E and F are
Lipschitz equivalent, then they are isomorphic.
Theorem 2.7 implies that when two “nice” spaces are Lipschitz equivalent, then each of them is isomorphic to a subspace of the other. The
following theorem, due to Heinrich and Mankiewicz 21, will show that under the additional assumption that E is complemented in E** they actually
embed as complemented subspaces of each other. This additional assump-
*
19
tion holds (trivially) when E is reflexive, and also when E is a dual space.
Theorem 3.2. Let E be a separable Banach space which is complemented
in its second dual and assume that F has RNP. If E and F are Lipschitz
equivalent, then E is isomorphic to a complemented subspace of F .
Thus, whenever E and F are such that we can also use the “decomposition
method” we deduce that the spaces are actually isomorphic.
Let us recall two cases where the decomposition method can be applied
and which cover “most” Banach spaces. We shall denote isomorphisms by
equality signs and assume that E = F @ W and F = E @ V .
(i) E and F are isomorphic to their squares.
Then
E = F @ W = ( F @ F ) @ W =F @ ( F @ W =
) F@E
and similarly F = F @ E , hence E = F .
(ii) There is a 1 5 p < 00 (or p = 0) such that E = ( E @.E@ . . . ) p .
Then
E = ( F @ W )@ ( F @ W )CB ... = F
@
(W @ F ) CB (W @ F )... = F @I3
and
F
=V
@E
=V
@( E @E @...) = ( V @E ) @ ( E @E
@
...) = F
@E
and again E = F .
As an immediate application we obtain that if E is Lipschitz equivalent
to L p ( p ) for some 1 < p < 03, then they are actually isomorphic. Indeed,
L p ( p ) is reflexive, hence by Theorem 2.7 E is isomorphic to a subspace of
L p ( p ) ,and therefore E is also reflexive. It follows from the theorem that E
and L p ( p )are isomorphic t o complemented subspaces of each other. Since
L p ( p ) satisfies condition (ii) above, the decomposition method applies.
We shall not prove Theorem 3.2. (For a proof see Corollary 7.7 in ’.)
We shall only prove the following earlier theorem of Lindenstrauss 28 which
illustrates how the condition that E is complemented in E** is used. The
proof of theorem 3.2 is obtained by combining the ideas in the proof this
theorem with those of the proof of Theorem 2.7.
Theorem 3.3. Let E be a Banach space and let EO be a closed subspace
of E such that there is a Lipschitz retraction f : E -+ Eo. Then there is a
bounded linear operator T : E + E,””such that T ~ =EidEo.
~ In particular,
if Eo is complemented in E,” (by a projection P ) , then EO is complemented
in E (by the projection Q = PT).
20
We shall present a proof due to Pelczyliski 36. We first need some
information on invariant means.
Let G be a group. Recall that a functional M E l,(G)* is called a
(left) invariant mean if it is a nonnegative functional such that M ( l ) = 1
and M ( f g )= M ( f ) for all g E G and f E 1.,
(We denoted the function
identically equal to 1 by 1, and fg is the left translation of f by g E G, i.e.,
fg(x) = f(gx).) Note that llMll = 1.
Proposition 3.3. Every abelian group G admits an invariant mean.
Proof. We first show that if W is a compact convex subset of a topological
vector space and if G is an abelian group of continuous affine transformations on W , then G has a common fixed point in W .
Choose a point w E W and fix g E G, then any limit point of
N-l
gj(w)is fixed by g. Thus the set Fg of fixed points of g , which is
compact and convex, is nonempty. Since G is abelian Fg is invariant under
any h E G and hence, by the same argument, h has a fixed point in Fg. It
follows that the fixed point sets {Fg : g E G} have the finite intersection
property, hence a nonempty intersection.
Put now W = {z* E 1,(G)* : x* 2. 0 a n d z * ( l ) = 1). Then W
is w*-compact and convex, and G acts on it as aEne transformations by
translation. Hence G has a fixed point in W , namely, a point M such that
M ( f g )= f for all g E G. Thus M is an invariant mean.
0
xr
We shall need vector-valued invariant means. Let E be a Banach space,
an operator M : lm(G;E) --t E is called an invariant mean if it is invariant
under translations, ((Ad((
= 1 and M ( f ) = z, where f denotes the element
in l,(G; E) which is identically equal to x.
Such vector-valued invariant means do not always exist, even when G
is abelian, and we may have to pass from E-valued means to means with
values in the larger space E**: if M I is a scalar-valued invariant mean on
G, then an E**-valued invariant mean M is given explicitly by the formula
(M(f),Y*) = Ml((Y*,f)) for all Y* E E*.
Proof of Theorem 3.3. Consider EOand E as abelian groups by using
their additive structure and fix two E:*-valued invariant means: one on
the Eo-valued bounded functions on the group G = EO and the other on
the Eo-valued bounded functions on the group G = E. To simplify the
notation we shall write them as integrals. This is a convenient notation
that makes it clear what are the fixed variables and what is the variable
21
with respect to which we average. No misunderstanding arises since we
do not use properties of the integral other than linearity and translation
invariance.
The first step is to average f in directions parallel to Eo. Put
E,**is Lipschitz with the same constant as f , it is the identity
Then g : E
on Eo, and it commutes with Eo-translations: if y E Eo, then
--f
=d z )
+ S(Y) = g(z> + Y
because the first “integral” is equal, by the invariance of the mean under
Eo-translations, to JEo ( f ( z x) - f ( x ) ) d x= g ( z ) .
We now define the operator T : E
E,* by
+
--f
Again T is Lipschitz with the same constant as f . The fact that g commutes
with EOtranslations implies that T is the identity on Eo,and a computation
0
similar to the above shows that T ( y z ) = T y T z for all y, z E E.
+
+
4. Uniformly Continuous Maps
Ultrapowers are an important tool in the study of the uniform classification
of Banach spaces. This was already demonstrated in Theorem 2.8. We start
with their definition and basic properties.
Let I be a set. A family U of subsets of I is called a filter if it is closed
under finite intersections, 0 @ U and B E U whenever A E U and B 2 A.
An ultrafilter is a maximal filter. An ultrafilter is free if nuA = 0.
Let X be a topological space, {xi}iE~
c X and U an ultrafilter on I .
We say that limuzi = x if the set {i : xi E 0 ) belongs to U for every
open neighborhood 0 of z. If X is a compact Hausdorff space, then every
{xi} converges with respect to any ultrafilter. (The limit may depend, of
course, on the ultrafilter.)
Let {Ei : i E I } be Banach spaces and let U be a free ultrafilter on I . Put
loo(Ei)= {x = (xi): zi E Ei and IIzlI = sup 11ziII < m}. The ultraproduct
of the Ei’s (with respect to U)is the quotient space (Ei)u = loo(Ei)/N,
where N is the closed linear subspace {x : limu llzill = 0) of loo(Ei)/N.The
22
norm on (Ei)u is given explicitly by llxll = limu IIzill and, by the definition
of N , is independent of the choice of the representing xi’s. When all the
Ei’s are equal to E we talk of the ultrapower of E and denote it by ( E ) u .
We identify E as a subspace of (E)u via the diagonal map x -+ (xi), where
xi = x for all i E I.
If Ti : Ei -+ Fi are uniformly bounded linear operators, then they induce
More
an operator (Ti)u : (Ei)u -+ (Fi)u by the formula (Ti)u(zi) = (Tixi).
generally, if fi : Ei 4 Fi are uniformly Lipschitz with sup (1 fi(0)II < 00,
then ( f i ) ~is defined similarly and is also Lipschitz.
Recall that a Banach space E is said to be finitely-representable in a
Banach space F if for every E > 0 and every finite-dimensional subspace
El of E there is a subspace F1 c F with d(E1, F I ) < 1 E . E is said to
be crudely finitely-representable in F if there is a constant C such that the
above holds with 1 E replaced by C.
We shall use the following “principle of local reflexivity” for ultrapowers:
For every Banach space E and for every free ultrafilter U the ultrapower
(E)u is finitely-representable in E . (This is analogous to the principle of
local reflexivity that says that for every Banach space E the second dual
E**is finitely-representable in E.)
We can now prove
+
+
Corollary 4.1. Assume that the separable Banach spaces E and F are
uniformly homeomorphic. Then E is isomorphic to a subspace of (F),**.
Proof. By Theorem 2.8 there is a Lipschitz embedding of E (which is a
subspace of (E)u) into (F)u. Considering the latter as a subspace of its
second dual (F);*, Theorem 3.1 implies that E is isomorphic to a subspace
of (F);*.
0
Note that the constants of the isomorphism are the same as the Lipschitz
constants of the equivalence given in Theorem 2.8, and that these constants
depend only on the moduli of continuity of the uniform homeomorphism
and of its inverse.
Proof of Theorem 2.9. Let f : E
F be the uniform homeomorphism
and fix a finite-dimensional subspace El of E. A standard back and forth
argument yields separable subspaces El c g c E and F^ c F such that
= F^: define inductively Fj = spanf(Ej) and Ej+l = spanf-lFj, and
then take and F^ to be the closures of U E j and U F j respectively. We can
thus assume that E and F were separable to start with.
--f
f(E)
23
By the corollary there is an isomorphism T : E -+ (F);*, with a constant which depends only on f. Fix C > IlTll //T-'Il. By local reflexivity
(between (F);* and ( F ) u and between ( F ) u and F ) there is an F1 c F
with d ( F l , T ( E l ) )as small as we wish, hence d ( E 1 , F l ) < C.
0
We now turn t o applications of approximate midpoint sets and we first
need to compute them in the various spaces. By translation we may do
the computation for the approximate midpoints between two points of the
form x and -x.
Lemma 4.1. (i) Let 1 5 p < 00 and let 0 # x E 1,. Then there is a finitedimensional subspace EO c 1, and a finite-codimensional subspace El c 1,
such that
1
-61/pIIxIIB(E1) c Mid(x, -x,6) c Eo (3p6)1/PllxllB(1,).
2
+
(aa) Let 0 # x E L1. Then there is an infinite sequence {xj} of exact
metric midpoints between x and -x with 11xi - xjll = 11x11 for i # j .
(iii) Let 2 < p < 00, and let 0 # x E L,. Then there is a constant C ,
depending only o n p , and a subspace F1 of L, of infinite codimension such
that
Mid(x, -x,6)
c F1 + C 6 1 / 2 ( ( x ( ( B ( L p ) .
Proof. By normalization we may assume that IIxlI = 1.
(i) Assume first that x is finitely supported, and choose N so that
x E EO= span{ej : j 5 N } . Put El = span{ej : j > N } .
If y E G1/PB(E1),then llxfyll = (IIxllP+ IlyllP)l/p 5 (1+6)l/P 5 1+6,
i.e., y E Mid(%,-x,6) .
For the other inclusion, assume that y satisfies 112 f yI( 5 1 + 6. Write
y = zo z1, where zi E Ei,and chose a sign 0 = f l so that llx 0zoII 2 1.
Then
+
+
+ 6IP L llx + @/(IP
Hence llzlll 5 ( ( 1 + 6 ) P - 1)"'
(1
+ BZOI(P + ( ( 2 1( ( PL 1 + ( I Z l ( l P .
= ((z
5 (2pb)'lP for 6 < 6(p).
If the support of x is not finite, we use the argument above for a finitely
supported vector which approximates x (where the degree of approximation
depends only on 6).
(ii) Inductively divide the interval [0,1] to 2j disjoint subsets {Ai,j :
i 5 2 j } such that Ai,j = Azi-l,j+l U A2i,j+1 and such that JAi,j1x1 = 2-j.
Then define z j ( t )= (--l)zx(t) for t E Ai,j and i = 1 , . . . , 2 j .
24
(iii) We only prove the case z = 1. The general case is similar but
requires some more technical details.
Let F 2 be the span of the Rademacher functions. By Khintchine’s inequality there is a constant A, such that ApllyllPI
((yI(2for every y € F2.
Take F1 t o be the closed subspace of L, orthogonal to span{ 1,F2}, i.e.,
F1
=
{z E L, : J z ( t ) ( l + y ( t ) ) d t = 0
Fix z E Mid(%,-x, 6) and write it as z
Assume, as we may, that a 2 0. Then
+ 6 2 111+
1
hence a
Zll,
+
2 111 zllz
=
5 6. By the same inequality
1+6 2
=
for every y E
a
F2
}
+ x i + x2 where xi E Fi.
+ + + 2 1+a
I
1 + 6 < 2, hence
l((1 a )
21
52112
11x2112
II(1+ a ) + 2 1 + 22112 2 111+ 22112
= (1
+ 11x211~)1/22 1 + Ilx2ll;/8 2 1 + A ~ l l ~ 2 l l ~ / 8
and 11x211p 5 (86/Ag)1/2.
Thus dist(z, F1) I ]la x2ll
+
I 6 + (86/A,)
2 1/2 .
0
We are now ready to sketch the proof of Theorem 2.10. We refer to
Theorem 10.13 in for details and for the proof for 1 < p < 2 which we do
not discuss here.
Proof of Theorem 2.10 for p = 1. Assume that f : L1 -+ I1 is a surjective uniform homeomorphism. Then f-’ satisfies a Lipschitz condition
for large distances and we choose a constant K so that JIy- zll 2 1 implies
K - l I l y - 41 I
Ilf(y) - f(z)ll. Fix 6 > 0.
By Lemma 4.1 the set Mid(%,-x, 0) is a “large” subset of L1 for every
x E L1: it contains an infinite IIxlI-separated set. We now apply Proposition
2.2. After translating the points to x and -x and assuming, as we may,
that llxll 2 1, we obtain for this x that this large set is supposed to be
mapped by f into Mid(f(z),f(-z),G), which is a “small” set in 11: it
is a finite-dimensional perturbation of a set of small diameter. But this
is impossible. Indeed, compactness of bounded sets in finite-dimensional
spaces would then imply that the images of two of the IIxlI-separated points
y, z E Mid(x, -x,O) would satisfy llf(y) - f ( z ) l l 6llxll - and for small 6
0
this would contradict llf(y) - f ( z ) l l 2 K-llly - zll = K-lIIx((.
-
Proof of Theorem 2.10 for p > 2. This case requires a new way to
measure “large” and “small” sets. We first give a heuristic formulation
25
of the so called "Gorelik principle", which uses a mixture of topological
and metric conditions t o compare the "size", and use it to give a heuristic
argument for the case p > 2. We then formulate a precise quantitative
version of the principle and apply it to give a precise proof.
The Gorelik principle: A uniform homeomorphism between two Banach spaces cannot take a large ball in a subspace of finite codimension in
one space into a small neighborhood of a subspace of infinite codimension
in the other.
Assume now that f : 1, -+ L, is a surjective uniform homeomorphism.
By Lemma 4.1 and Proposition 2.2 f should map a Sl/p((zl(-ballin a subspace El of finite codimension in 1, into a 61/211z11-neighborhoodof an
infinite-codimensional subspace F1 of L,. But this would contradict the
Gorelik principle because p > 2 implies that for small S the radius of the
ball, bl/PIIzII, is much larger then the size, S 1 ~ 2 ~of~the
z ~ neighborhood.
~,
We now give the precise formulation of Gorelik's theorem
19.
Theorem 4.1. Let E and F be Banach spaces, and assume that 'p : E -+ F
is a homeomorphism with a uniformly continuous inverse 'p-l. Assume
that El c E is a subspace of finite codimension and Fl c F a subspace of
infinite codimension. If a ,P > 0 satisfy
+
q(aB(E1))c F1 P B ( F ) = { z E F : dist(z, F1) I P }
(*)
then the modulus of continuity of 'p-l satisfies w,-i(2P) 2 a/4.
To deduce the case p > 2 of Theorem 2.10 we use the fact that f-'
satisfies a Lipschitz condition for large distances to choose K > 0 so that
t 2 1 implies w f - l ( t ) 5 K t . Fix S > 0 and use Proposition 2.2 and Lemma
4.1 to choose z with 261/211z11 2 1 so that f maps a 61/pIlzll-ball in a
subspace El of finite codimension in I , into a 61/211zll-neighborhoodof an
infinite-codimensional subspace Fl of L,. Theorem 4.1 with a = S1/p((z((
and P = 61/211z11then gives
w f - l ( 2 6 1 ~ 2 ~ ~2z 6~ 1~ )~ q x ~ =
~ f51/p-1/2
/4
. N 2llxll/4
But wf-l (261/211z11) I K.261/211z11by the choice of K , and this is impossible
for small enough 6 (because p > 2).
Proof of Theorem 4.1. The proof will follow from two facts:
(i) There is a compact set A c ; B ( E ) such that whenever f : A -+ E
is a continuous function satisfying 11 f (z) - 211 1. a/4 f o r every z E A , then
f(A) n Ei # 0.
26
Indeed, by a theorem of Bartle and Graves (see Proposition 1.19(ii) in
) there is a (nonlinear) continuous right-inverse : E/E1 -+ E to the
quotient map IT : E 4 E/E1 so that maps ? B ( E / E l ) into $ B ( E ) . The
set A = + ( F B ( E / E 1 ) )is compact because E/E1 is finite-dimensional,
If f : A 4 E satisfies Ilf(z)- 211 I a/4 for every z E A, then the map
g(y) = y - ~ f ( + ( y ) ) maps %-B(E/E1) into itself because ~ $ J ( y v =
) y,
1 1 1 ~ 1 1 = 1 and $(y) E A yield
+
+
By Brouwer's fixed point theorem g has a fixed point yo, and it follows that
Tf(+(YO)) = 0, i.e.1 f(G(Y0)) E f ( A )n El.
(ii) Let B be a compact subset of F . Then there is a point y E F with
IIyyII < 2P and dist(B
+ y, F I )> P.
Indeed, let B1 be a finite PIZdense set in B . The subspace G spanned
by F1 and B1 is a proper subspace of F (because F1 has infinite codimension), hence there is a point y E F with llyll < 2P and dist(y,G) > 3 p / 2 .
Given z E B , choose z1 E B1 with llz - zlll < P / 2 . If z E F1, then
z1 - z E GI hence JIz y - zll 2 dist(y, G) - llz - z111 > p.
+
To prove the theorem, find a compact set A C ; B ( E ) as in (i). Then
use (ii) for the compact set B = cp(A), and find y E F with llyll < 2,B for
which dist(cp(A) y, F1) > P.
The map f ( z ) = cp-'((p(s) y) maps A into E , and if we had
w,-1(2p) < a / 4 , then f would satisfy
+
+
Ilf(z) - zll
=
Ilv-l(cp(4+ Y)
-
cp-'(cp(4)II
I W,-1(IIYII> 5 W,-1(2P) < 4 4
for every z E A. By (i) there is an zo E A with f ( z 0 ) E El. In fact
~ ( x o E) aB(E1) because I l f ( ~ o ) l l I Ilf(zo)- z o l l + llzoll < 3 a / 4 < a.
Condition (*) then implies that cp(zo) y = cp(f(z0)) E FI PB(F),
contradicting dist(cp(A) y, F I ) > ,B.
+
+
+
We finish by quoting three results on spaces whose linear structure is
uniquely determined, or is determined up to finitely many possibilities, by
their metric structure. The proofs use the techniques of this section as well
as quite deep results in the linear theory of Banach spaces. Parts (i) and
(iii) are due to Johnson, Lindenstrauss and Schechtman 2 5 , part (ii) is due
to Godefroy, Kalton and Lancien 18.
27
< 00. If E i s uniformly homeomorphic t o I,,
t h e n it i s isomorphic t o it.
(ii) If E i s Lipschitz equivalent t o Q, then it is isomorphic t o it.
(iii) For every n there i s a Banach space whose u n i f o r m class contains
exactly 2n different linear structures. More precisely, there are 2n mutually nonisomorphic Banach spaces E l , ...,En, which are uniformly homeomorphic t o each other, and so that a n y Banach space which i s uniformly
homeomorphic t o t h e m i s isomorphic t o one of the Ei 's.
Theorem 4.2. (i) Let 1 < p
It is unknown whether the analog of (i) also holds for L, (when p # 2).
In other words, it is unknown whether a Banach space which is uniformly
homeomorphic to L, is necessarily isomorphic to it. It is also unknown
whether (i) holds for 11. In fact, it is not even known whether a Banach
space which is Lipschitz equivalent to 11 is necessarily isomorphic to it. (The
point is that it is unknown whether a space which is Lipschitz equivalent to
a dual space is necessarily isomorphic to a dual space. If E is a dual space
which is Lipschitz equivalent to 11, then it is isomorphic to it by Theorem
3.2 and the decomposition method.)
It is unknown whether part (ii) holds for spaces which are uniformly
homeomorphic t o CO.
In part (iii) one can take El = T ( p l ) @ ... @ T ( P n ) , where T ( p ) is the
pconvexified Tsirelson space, and 1 < p l , ...,pn < 00 are different from
each other. The 2" mutually nonisomorphic but uniformly homeomorphic
spaces are El @ ( CjEJ
@ l p j ) , where J is an any subset of (1, ...,n } .
It is not known how t o construct a space with k different linear structures when k is not a power of 2.
For proofs see also Theorem 10.15 (for (i)), Theorem 10.17 (for (ii))
and Proposition 10.40 (for (iii)).
*
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(1994), 1-8.
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31
AN INTRODUCTION TO LOCAL DUALITY
FOR BANACH SPACES *
MANUEL GONZALEZ
Departamento de Matemdticas
Facultad de Ciencias
Universidad de Cantabria
39071 Santander (Spain)
E-mail: [email protected]
We present an introduction to the study of the local dual spaces of a Banach
space. We describe with detail the main properties of this concept and give several
characterizations. These characterizations allow us to show examples of local dual
spaces for many classical spaces of sequences or functions.
Introduction.
A (closed) subspace 2 of the dual of a Banach space X is a local dual
of X if for every couple of finite dimensional subspaces F of X* and G of
X, and every 0 < E < 1, there is an operator L : F
Z satisfying:
-
(a)(1 - E)Ilfll 5
(b)
IlLfll I (1 + E ) I l f l l , Yf E F ;
Wf,4 = (f,4, Yx E G, Yf E F ;
(c) L ( f ) = f, b'f E F
n 2.
A trivial example is given by 2 = X*.
The PRINCIPLE OF LOCAL REFLEXIVITY (P.L.R., FOR SHORT) gives a
nontrivial example. This principle can be stated by saying that X ,as a
subspace of the second dual X * * , is a local dual of X*.
One reason to study local dual spaces of X is that in many cases X* is
a big space and for we do not have a good representation of X*.
In this paper we present an introduction to the study of local dual
spaces of Banach spaces. Our aim is to help to understand this concept.
*Work partially supported by the DGI (Spain). Proyecto BFM2001-1147.
32
So from the beginning we have explained with some detail the meaning
of the conditions that appear in the definition of local dual. We have
included many examples of local dual spaces, some of them elementary,
and some other that could be interesting because they are far more simple
that the whole dual space. We have also given several characterizations
of the subspaces of X* which are local duals of X and some other results
which provide additional properties or techniques to find examples of local
duals.
We have tried to write the paper in such a way that could be read
by people that have attended a course of functional analysis. We have
included some proofs that are not too complicated and help to understand
the concept of local dual. Since many of these proofs need the concept of
convergence over an ultrafilter, we have included at the end of the paper
a short appendix describing this concept. This should be enough to follow
the proofs.
Our material is organized into eight short sections plus an appendix.
In the first section we analyze the properties that define the concept of
local dual space. The second section contain a characterization in terms
of finite dimensional subspaces, which is a weakening of the definition of
local dual, and some characterizations of “global” type, in which only the
whole spaces appear. These global characterizations are easier to deal with
than the definition. However there is a disadvantage: the second dual X**
appears in the statements. In the third section we show that being a local
dual is a symmetric relation: 2 is a local dual of X if and only if 2 has
a local d u d isometric to X . Later in the paper we will apply many times
this property and the characterizations given in Section 2 to find examples
of local dual spaces.
In Section 4 we give some natural examples: C[0,1]is a local dual of
L1[0,1] and the Bore1 measures on [0,1] which are absolutely continuous
with respect to the Lebesgue measure form a local dual of C[O,11 isometric
to L1[0,1]. We also give some examples of local duals for the spaces e , ( X )
and &(X) of vector sequences and for the spaces C ( K , X ) ,L1(pL,X) and
L,(p,X) of vector-valued functions. In Section 5 we show local duals
for spaces with a basis. Section 6 contains some additional properties of
local dual spaces. Every separable space admits a separable local dual and
some spaces admit a smallest local dual. We have included these properties
so late in the paper in order to have at our disposal examples to illustrate
them. Section 7 contains local dual spaces for projective and injective tensor
33
products of Banach spaces. Note that the spaces C ( K , X ) and L1(p,X)
are tensor products 5 .
In the eighth section we introduce the concept of ultrapower of Banach spaces. The P.L.R. for ultrapowers, due to Heinrich 15, give us an
interesting example of local dual. Moreover, we give characterizations of
the subspaces of X * which are local duals of X in terms of ultrapowers.
Finally we give an appendix on ultrafilters. It contain the definition of convergence over an ultrafilter and the fact that the ultrafilters in a compact
set converge. This result is an essential tool in the previous sections.
Throughout the paper we use standard notations: X and Y are Banach
spaces, Bx the closed unit ball of X , and X * the dual space of X . Subspaces
are always closed. We identify X with a subspace of X**.
For A c X we consider the set
A I : = { f ~ X *. . (,z)
f = 0 for every z E A}.
Analogously, for C c X * we define the set C_Lof X . We denote by B ( X ,Y )
the space of all (bounded linear) operators from X into Y , and by K ( X ,Y )
the subspace of all compact operators. Given T E B(X,Y ) ,N ( T )and R(T)
are the kernel and the range of T , and T* is the conjugate operator of T .
We denote by W the set of all positive integers.
WARNING: Sometimes we say that some concrete space Z is a local dual
of X without specifying the position of Z inside X . In these cases it should
be clear to what copy of Z we are referring to. A simple example is C[O,11
is a local dual of L1[0,1]. A not so simple example: L1[0,1] is a local dual
of C[O,11. For the latter case see Example 3.
The concept of local dual space has been introduced recently 11, but
there are some related concepts that have been previously studied by many
authors. It would have been difficult to acknowledge all the relations with
previous results. So we have mentioned some of them throughout the paper
and we mention here some of the authors that have studied these properties:
Jerry Johnson 16, Stefan Heinrich 15, Nigel Kalton 18, Gilles Godefroy 7 ,
Asvald Lima l9 and Santiago Diaz '.
Many of the results we present here are joint work of Antonio Martinezl2 13.
Abej6n and the author in
I am grateful to Antonio Aizpuru and Fernando Le6n-Saavedra for their
invitation and for their effort to create a pleasant atmosphere for all the
people which attended the first international course of mathematical analysis in Andalucia.
'' ''
34
1. Basic properties
Here we study the properties of the local dual spaces of a Banach space
which are a direct consequence of the definition.
Definition 1. A subspace 2 of the dual of a Banach space X is a local
dual of X if for every couple of finite dimensional subspaces F of X* and
G of X and every 0 < E < 1, there is an operator L : F
2 satisfying
-
( a ) (1 - &)llfll
5 IlLfll I (1+ &)llfll, Yf E F ;
( b ) ( L f , z )= (f,z),Yx E G, Vf E F ;
( c ) L ( f )= f, Vf E F n 2.
Clearly the dual space X * is a local dual of
X.
Example 1. PRINCIPLE OF LOCAL REFLEXIVITY
of X**,
is a local dual of X * .
For a short enlightening proof of the P.L.R. see
(I7):
x as a subspace
2o
Remark 2. Let 2 be a local dual of X.
(1) Property ( a ) says that X* is finitely represented in 2: the isometric properties of each finite dimensional subspace F of X* are arbitrarily close to the isometric properties of some finite dimensional
subspaces L ( F ) of X .
( 2 ) Property ( b ) says that L preserves the duality when we pass from
the pair (G, F ) to the pair (G, L ( F ) ) .
(3) The meaning of property (c) is more difficult to grasp. It says that
the operators L fix points of 2,and it is related with the fact that
'
2 is complemented in X**.
See Theorem 5.
Exercise 1. Let 2 be a local dual of
X.
(1) The subspace 2 is norming. This means that for every z E X ,
II4I = SUP{I(Z, .)I
(2) Let z** E 2
'
.
: z E 2 ; llzll = 1).
Then
llz** - 211 2 llzll for all z E X .
Clearly, X* is the only local dual of X when X is reflexive. Moreover,
this can also be true for some non-reflexive spaces.
35
Exercise 2. Apply part (2) of Exercise 1 to show that
dual of q,.
C1
is the only local
NOTES AND REMARKS
(1) Being a local dual of X is an isometric property and depends on the
position of 2 inside X * . It is not preserved under renormings of X.
(2) Two local dual spaces for a Banach space can be very different from
an isomorphic point of view. Indeed, for the space el the copy of
Q inside loo is a local dual of X , and also each predual of el is
isometric to a subspace of loo which is a local dual of C1. However,
there are preduals of C1 which do not contain subspaces isomorphic
to CO. See Bourgain’s spaces in ’.
2. Some characterizations
Here we give some characterizations of the subspaces of X* which are local
dual spaces of X . These characterizations are useful to understand the
concept of local dual.
The first characterization should be compared with the definition. Here
we replace the “exact” conditions (b) and (c) by the “approximate” conditions (b’) and (c’). As we shall see, this is very useful to simplify some
arguments.
Theorem 3. (Local Characterization) A subspace Z of X* is a local dual
of X if and only if f o r every couple of finite dimensional subspaces F of
X* and G of X , and e v e y 0 < E < 1, there is a n operator L : F
2
satisfying
-
PROOF.
A direct proof can be done using an argument similar to the one
in part (b) of Proposition 2 in
proof of Theorem 2.5 in ll.
13.
For an indirect argument we refer to the
0
Example 2. We can apply Theorem 3 to show directly that co is a local
dual of .!?I
36
-
Let Pn : em
Q be the projections defined by Pnek = ek for k I n
and Pnek = 0 for k > n.
It is immediate to check that these projections satisfy the following
properties.
(1) llPnll = 1 and IIPnzIIm 4 llzlloo for all z E em.
( 2 ) llPnZ - ~
1 + O1 for~ all z E Q.
(3) (PnZ,y) 4 ( z , y ) for all z E loo
and all y E e l .
Let F c C,
G E el and E > 0.
CLAIM:There exists an integer n such that Pn satisfies the properties of
the map L associated to F , G and E in Theorem 3.
Indeed, taking into account that the unit ball of a finite dimensional
space is compact, (a) follows from (l),(b’) follows from (2) and (c’) follows
from (3).
Although the definition of local dual space is “local”, we shall see in
Theorem 5 that it is possible to give “global” characterizations. The following result will be useful.
Proposition 4. Let 2 be a local dual of X . Then every compact operator
T : 2 --+ Y admits a compact extension T : X* Y with
= IlTll.
I]?]
--+
PROOF.Let A be the set of all the pairs CY = ( F ,G) of finite dimensional
subspaces F c X * and G c X . For each CY = (F,G) E A, we denote by L,
the map from F into 2 associated to F , G and 6 = (dim E, dim Fa)-’
in the definition of local dual.
Note that for every f in the unit ball of F and every CY E A, TLaf is
contained in the compact set 2T(Bz). Thus taking an ultrafilter U on A
refining the filter associated to the inclusion, we can define T : X * 4 Y as
in Remark 29
+
Tf
:=
lim TL,f.
Ol-U
It is enough to check that this operator satisfies the required conditions. 0
Exercise 3. Complete the proof of Proposition 4.
-
Let Y be a subspace of X. An operator x : Y *
X* is an extension
operator if ( x f ) I Y = f for every f E Y*.Observe that 11 f 11 5 Ilxfll.
The operator x is an isometric extension operator if IIx f I( = llfll for
every f E Y*.
37
Exercise 4. Let Y be a subspace of X and let J : Y + X denote the
embedding of Y into X .
Show that an operator x : Y*
X * is an extension operator if and
only if J * x is the identity on Y * .
In this case, x J * is a projection from X * onto x ( Y * ) .
-
In the following result we give “global” characterizations of the property of being a dual local. These characterizations are easier to deal with
than the definition. However there is a disadvantage: the second dual X**
appears in the statements.
Theorem 5. (Global characterizations) For a subspace Z of X * , the fol-
lowing statements are equivalent:
-
(1) Z is a local dual of X .
(2) there is an isometric extension operator x : Z*
R(X) 3
(3) there exists a norm-one projection P : X**
N ( P ) = Z* and R ( P ) 2 X ;
(4) there exists a norm-one projection Q : X***
R(Q) = 2’’ and N ( Q ) c X I .
x;
PROOF.
Let
L
: 2 -+
X** so that
X** such that
X*** such that
X * denote the inclusion.
(1) =+ (2)
For every finite dimensional subspace H of Z* we consider the quotient
map q H : 2 -+ Z/H*. By Proposition 4 there exists an extension QH :
X * -+ Z/HL of QH with 1 1 Q ~ l l = 114Hll.
Since ( q H ) * is the embedding of H into Z * , X H := ( Q H ) *: H 4 X** is
an isometric extension operator. Indeed, llx~ll= ( ( q h ( =
( 1. Moreover, let
h E H and z E 2.
(XHh1 2) = ( h1 Q H Z ) = ( h 1 q H z ) = ( (qH)*h> 2) = ( h z ) .
Now we take a nontrivial ultrafilter V on the set of all finite dimensional
subspaces of H of Z* refining the filter associated to the inclusion. Since
Bx.. is weak*-compactl we can define a map x : Z* -+ X** by
x := weak*-H-V
lim
XH.
Clearly x is an isometric extension operator. It remains to check that
X ( Z * )3
Let x E X . Then L * X E Z* and for each f E X * ,
x.
38
Thus x satisfies the required conditions.
(3) Let x be the operator given in (2). The kernel of the conjugate
(2)
operator L* : X** + Z*is ZL, and L*X is the identity on Z*.Therefore
XL* is a norm-one projection on X * * , N ( x L * )= N ( L * )= ZL and R ( x L * =
)
R(X) 3
(3)
x.
+ (4) Take Q = P*.
(4)
ivity to
details.
(1) It is essentially an application of the principle of local reflexfor
Z c Z**= 2". We refer to the proof of Theorem 2.5 in
''
0
Remark 6. Recall that X is a local dual of X * by the P.L.R. In this case
the decomposition given by the projection in the third part of Theorem 5
is
x***
=X*@XL,
and the projection P on X*** is just the operator restriction to X .
Remark 7. The range of the projection P which appears in part (3) of
Theorem 5 is weak*-dense because R ( P ) 3 X . Therefore, in the nontrivial
case Z # X * there is no subspace M of X * such that R ( P ) = M I .
NOTES AND REMARKS
If in the definition of local dual we delete property (c), then we obtain
the concept of finite dual representability (f.d.r.) studied in lo. There are
examples of Banach spaces X and subspaces Z of X * such that X * is f.d.r.
in 2,but 2 is not a local dual of X . Note that clearly 21 c 2 2 and X *
f.d.r. in 21 implies X*f.d.r. in 22.However,'
2 is complemented when Z
is a local dual, and 2
: complemented does not imply 2; complemented.
See Example 2.11 in
3. Symmetry
Here we shall show that being a local dual space is a symmetric property in
some sense. This fact is a useful tool to find examples of local dual spaces
for some concrete Banach spaces, as we shall see throughout the paper.
Let Z be a local dual of X. Let L : Z -+ X* and J : X
embedding maps. We introduce the map
T = L * J :X -+ Z*.
-+ X**
be the
39
Since
Z is norming (Remark 1)]T is an isometry.
Theorem 8. Let Z be a local dual of X. Then T(X) is a local dual of Z
isometric to X.
-
PROOF.
Let x : Z*
X** be an isometric extension operator such that
R ( x ) 3 X . Then L*X is the identity on Z*and XL* is a projection on X**
with R(xL*)= R(x) 3 J(X). Therefore, for every x E X I xTz = XL*JX
=
J x.
CLAIM:The map $J : T(X)* --+ Z** defined by
($Jf,d:= (XSI f
O
f E T(X)*I 9 E Z*l
T);
is an isometric extension operator and R(+) II 2.
Clearly
ll$Jll 5 1. Moreover] for every f
E T(X)*and every Tx E T(X),
($f,Tz) = (XTZ,f 0 T) = ( J x f 0 T) = (f 0 T, x) = (f]Tx).
]
Thus the claim is proved.
Let y E 2. We can write y
=f o
(+fl9) = (xg1 f
for every g E Z * ; hence
O
TI for some f 6 T(X)*.Then
T) = (xg1 Y) = (91Y),
R(+)II 2.
0
4. Some natural examples
The theorem of representation of Riesz (Theorem 6.19 in 2') allows us to
identify the dual space of the space C[O,11 of the continuous functions on the
unit interval [O,11 with the space M[O,11 of all the Bore1 measures on [0,1].
More precisely, for each F E C[O,1]*there exists a meamre p~ E M[O,11
so that
for every g E C[O,11. Moreover, the map F 4 p~ is a bijective isometry.
By the Radon-Nikodym theorem (Theorem 6.10 in 21)1 every p E M[O,11
can be decomposed into two parts p = pa p s 1where pa and p, are absolutely continuous and singular, respectively, with respect to the Lebesgue
measure on [0,1]. Moreover] JJpJJ
= IIpaJJIlpsJ)
and there exists a function
f E L1[0,1] such that dpa = f dt and \\pall = IJflll.
+
+
40
In this way we obtain a decomposition
M[O,11 3 -Wzc[O, 11 el Msing[O, 11
Example 3. M,,[O, 11 is a local dual of C[O,11 isometric to L1[0,1].
PROOF.
Along this proof we identify M,,[O, 11 and L1[0,1].
For every integer n we consider the partition of [0,1] given by the intervals
[(i - 1)/2", i/Y) if i = 1,. . .2" - 1,
I? =
[(2, - 1)/2,, 11 if i = 2,.
{
xF
Let us denote by
the characteristic function of I?. We define an operator
G, : M[O,11 -+ L1[0,1] by
i=l
Then we check that the maps G, satisfy
(1) Each G, is a norm-one projection.
(2) The sequence (Gnf) converges in norm to f, for every f E L1[0,1].
(3) (G,X) converges to X in the weak*-topology and ))G,XII 4 IlXll, for
every X E M[O,11.
Finally we proceed as in the proof of Example 2.
0
Remark 9. It is possible to give an alternative proof of the fact that
M,,[O, 11is a local dual of C[O,11taking limits with respect to an ultrafilter.
We define the map
x: M,,[O,
by
x h := weak*-lim G;
1]*= L,[O, 11
-
M[O,1]*
h.
Clearly llxll 5 1. So it is enough to show that x is an extension operator
satisfying R(x) 3 C[O,11.
Let h E L1[0,1]* and f E L1[0,1].
(xh, f ) = lim(G:
Thus x h is an extension of h.
h, f ) = lim(h, G, f ) = (h, f).
41
Let g E C[O,11 and p E M[O,11.
0
Hence x g = g and R(x) 3 C[O,11.
Example 4. C[O,11 c L,[O, 11 is a local dual of Li[O, 11.
PROOF.
We have just seen that M,,[O, I] e L1[0,1] is a local dual of C[O,11.
Thus applying Theorem 8, it is enough to show that T(C[O,11) is the copy
of C[O,11 in L,[O, 11.
Let L : L1[0,11 = M,,[O, 11 -+ M[O,11 and J : C[O,11 -+ C[O,1]**be the
inclusion maps. Recall that T = L*J.
Let g E C[O,11. For every f E L1[0,1]
(L*JS, f) =
Jd
1
(L*Jg)(t)f(t)dt
and
( L * J S , f ) = ( J g , L f ) = (Lf79) =
Thus ( L * J g ) ( t ) = g ( t ) a.e. Hence
L*
J g = g.
I'
g(t)f(t)dt.
0
Remark 10. Both spaces L1[0,1] and C[O,11 admit a separable local dual,
although their dual spaces are nonseparable. We will see later that this is
a consequence of a general result: Each separable Banach space admits a
separable local dual.
Example 5.
(1) l , ( X * ) is a local dual of l,(X).
(2) & ( X ) is a local dual of el@*).
First case: For every couple a := (E,F ) of finite dimensional subspaces
of ll(X*) and l,(X**), we select a pair of sequences of finite dimensional
subspaces (En) in X * and (F,) in X** so that E c ll(En)and F c l,(Fn).
We denote la1 := dim(E) dim(F).
The principIe of local reflexivity allows us to find, for every n, an [a[-'isometry SE : Fn
X so that ( S z f , e ) = ( e , f ) for every e E En and
f E F,, and S;(f) = f for every f E F, n X . We consider the map
S" : F -+ l,(X) given by S"(zn) := ( S z ( Z n ) ) .
-
+
42
Let U be an ultrafilter in the set of all couples a = ( E , F ) of finite
dimensional subspaces E of l l ( X * ) and F of l,(X**) refining the order
filter. We define an operator A : l,(X**)
l,(X)** by
-
A(a,) :=weak*- lim S 0 ( t n ) .
a-U
Note that A is an isometry and A(y,) = (y,) for every (y,) E l,(X**).
Therefore, A is an isometric extension operator. Moreover, A((zn)) = (z,)~
if (z,) E C,(X). In particular A(l,(X**)) 2 l,(X).
Second case: Observe that the operator T : l,(X)
duced in Section 3, is the natural inclusion.
-
l , ( X * ) * intro-
The first part of the previous example is particularly useful because the
dual space of & ( X ) is a "big" space and does not admit a good representation. This is also the case for the spaces considered in the next example.
Example 6. (See
and
12)
Let p be a finite measure.
(1) L l ( p , X * ) is a local dual of L m ( p , X ) .
(2) L,(p, X * ) is a local dual of L1(p,X ) .
NOTESAND REMARKS
From
it follows
M[O,11" = Mac[O,111
@
,
MsingI0, 111
However, this fact does not allow us to apply Theorem 5 to derive that
Ma,[O,11 is a local dual of C[O,11. Indeed, Msing[O,11' does not contain
C[O111.
5. Spaces with a basis
Recall that a sequence ( e n ) in a Banach space X is a (Schauder) basis of
X if for every z E X there is a unique sequence (a,) of scalars so that
x = C z l aiei.
Let (e,) be a basis of X . The projections P, : X -+ X defined by
P,(CeO az. ea. ) := Cy=laiei are continuous.
The basis (en) is monotone if llPnll = 1 for every n.
43
The linear functionals e: defined by e : ( x & aiei) = a, are continuous.
We denote by [e;] the closed subspace of X * generated by {e; n E W}.
Theorem 11. Let X be a Banach space with a monotone Schauder basis
(e,). T h e n [e:] is a local dual of X .
PROOF.Let U be an ultrafilter on W. Since the unit ball of X** is weak*compact, we can define a map P on X** by
Pz :=weak*- lim P;*z,
n+U
z E X**.
Note that PnPk = PkP, = P, for n 5 k. Thus P,**P = PP;* = P, for
every n. Since each P, is a norm one projection, P is a norm one projection
on X**.Also it is clear that R ( P ) I X .
Now N ( P ) c N(P,) for every n. Thus N ( P ) c nr="=,N(P;*). But
the sequence of subspaces (R(P,*))is increasing and U&R(P,*) is dense in
0
[e:]. Therefore [e:]' = n&N(P,**) c N ( P ) . Thus [e:]l = N ( P ) .
The following example is similar to those provided by Theorem 11.
Example 7. The subspace 2 of Lm[O,11 generated by the characteristic
functions xn,i of the dyadic intervals
is a local dual of L1[0,1].
To prove it we consider the sequence (P,) of projections defined by
2n
Pnf := C(2nXn,i,
f)Xn,i,
i=l
and repeat the argument in the proof of Theorem 11. We refer to l 1 for
more details.
It is not difficult to show that this subspace 2 is isometric to the space
C(A) of the continuous functions on the Cantor set A in [0,1].
NOTESAND REMARKS
It was proved by Casazza and Kalton that a separable Banach space
X has the M.A.P. if and only if we can find a commuting approximating
sequence in X ; i.e., a sequence of finite rank operators T, acting on X so
that
lim IIT,a: - a:((= 0 for all 5 E X ,
n+m
44
(b)
lim llTnll = 1 and
n-co
( c ) TnTk = TkTn = Tmin{k,n);
Using this fact we can prove the following result.
11) Let X be a separable Banach space
with the M.A.P., and let (T,) be a commuting 1-approximating sequence o n
X . Then U,"==, R(T,*)is a local dual of X and has the M.A.P.
Theorem 12. (Theorem 2.15 in
Remark 13. Let X be a space with a monotone basis. Then the projections Pn : X + X form a commuting 1-approximating sequence on X .
6. Further properties
The first result in this section extends Theorem 11. It is essentially a special
case of Lemma 111.4.3 in 1 4 . Its proof in l4 is based in some ideas contained
in ".
Theorem 14. Every separable Banach space admits a separable local dual
space.
The space L1[0,1] does not admit a smallest local dual; i.e., there exists
no local dual z d contained in every local dual of L1 [o, 11.
Indeed, we have seen in the previous section that L1[0,1] admits two
local dual spaces C[O,11 and C(A). These subspaces of L,[O, 11 have empty
intersections.
Here we shall see that this cannot happen to spaces that contain no
copies of C,.
Theorem 15. If
X contains no copies of el, then it admits a smallest local
dual.
For dual spaces we have a similar result, which is an application of
Proposition v.1 in 7.
Proposition 16. (Proposition 2.22 in 11) Suppose that X i s isometric t o a
dual space. Then X admits a smallest local dual z d i f and only i f it admits
a smallest normang subspace 2,. I n this case z d = Zn, and this space is
the unique isometric predual of X .
NOTESAND REMARKS
Let dens(X) stand for the density characterof X, defined as the smallest
cardinal K for which X has a dense subset of cardinality K .
45
The following result gives an extension of Theorem 14. It is essentially
a special case of Lemma 111.4.4 in 14.
Proposition 17. Every subspace L of X* is contained in a local dual ZL
of X with dens(2L) = max{dens(l), dens(X)}.
7. Tensor products
Here we describe some local dual spaces of injective or projective tensor
products of Banach spaces. For information on tensor products of Banach
spaces we refer to 5 .
Let X and Y be Banach spaces. Let B(X; Y) denote the vector space of
all bilinear maps on X x Y and let B(X; Y)* denote the space of all linear
functionals on B ( X ;Y).
For each pair z E X and y E Y we define z @ y E B(X;Y)* by
(z @ y
, A ) := A ( x ,y).
Definition 18. The tensor product X @ Y of the spaces X and Y is the
subspace of B(X; Y)* generated by {z @ y : z E X, y E Y}.
From the norms on X and Y we can derive many norms on X @ Y. The
most popular ones are the projective norm 11 . llT and the injective norm
II . I l e .
Let a E X@Y. Observe that the representation a = zl@yl+. . *+zn@yn
is not unique in general, e.g., (x y) @ z = z z y @ z .
We define the projective norm 1) . \IT on X @ Y by
+
+
Definition 19. The projective tensor product X&Y of X and Y is the
completion of (X Y , 11 . llT).
We can identify (XG,,Y)* with the space B(X, Y*) of all the operators
from X into Y* by defining
( T , z @ d= (Tz,y).
In the previous section we considered the M.A.P. for separable Banach
spaces. In general, we say that a Banach space X has the M.A.P if for
every E > 0 and every compact set K in X ,there is a finite rank operator
T on X such that llTll 5 1 and IITz - 211 5 E for every x E K .
46
Most of the classical Banach spaces have the M.A.P.; for example, the
space L 1 ( p ) of integrable functions with respect to a finite measure p, the
space C ( K )of continuous functions on a compact space K and its respective
dual spaces have the M.A.P.
The following result is proved using some ideas of 16.
Theorem 20. Suppose that Y * has the M.A.P.T h e n the subspace
K ( X , Y * ) of the compact operators in B ( X , Y * ) is a local dual of X G j , Y .
PROOF.Since Y * has the M.A.P. there exists a net (A,) of finite rank
operators on Y* with IIA,II 5 1, so that lim, llA,g - 911 = 0 for every
g E Y * . Note that (by compactness) we can assume that (A,) is weak*convergent in K(Y * ) **.
Now, given T E B ( X , Y * ) and @ E I c ( X , Y * ) * ,the expression
( @ T ,A ) :=
(a, AT)
-
defines @T E K ( Y * ) * .Thus we can define A : K ( X ,Y*)*
(A@,T ) := lim(@,A,T)
a
B ( X ,Y * ) *by
= lim(A,, @ T ) .
Q
Note that for every f @ g E X * @ Y* we have
(A@,f G3 9 ) = l ai d @ ,Aa(9) . f) = (@,
f @ 9).
So A is an isometric extension operator. Analogously, we can check that
for every z 8 y E X&,Y c B ( X , Y * ) * ,we have
A(. @ Y I K ( X , Y * ) ) = 5 @ Y.
Thus XG,Y c A ( K ( X ,Y * ) * )and
, it is enough to apply Theorem 5.
0
NOTES AND REMARKS
We can define injective tensor product X & Y of X and Y as the completion of ( X @ Y , 11 . / I E ) , and obtain some results similar to those given
for projective tensor products.
We observe that the description of ( X & Y ) * is more complicated. It
can be identified with the space Z ( X ,Y * )of all the integral operators from
X into Y * with its natural norm.
Theorem 21.
dual of X & Y .
Suppose that Y*has the M.A.P.T h e n X * & Y * i s a local
Moreover, we can identify the space L l ( p , X ) of vector-valued integrable
functions with X&L1 (p)and the space C ( K ,X ) of continuous vector Valued functions with X & C ( K ) 5 . From these facts, taking into account that
47
L 1 ( p ) *and L , ( p ) are C ( K )spaces and that C ( K ) *is a & ( p ) space, we
obtain additional examples.
8. Ultrapowers of Banach spaces
The ultrapowers of Banach spaces allow us to deal with some properties of
a Banach space X defined in terms of finite dimensional subspaces, i.e., the
local properties. Many times a local property of X corresponds to a global
property of an ultrapower XU of X with respect to some ultrafilter U .We
refer to l5 for a good survey on ultrapowers of Banach spaces. See also
'.
Here we consider the ultrapowers of Banach spaces to give new examples
of local dual spaces and to present some characterizations of the subspaces
of X* which are local dual spaces of X .
We saw in part (a) of Remark 2 that X* is finitely represented in each
local dual 2 of X . This is equivalent to say that there exists an ultrapower
ZU of Z such that X* is isometric to a subspace of ZU.
In order to define the ultrapowers of Banach space we fix an infinite set
I and a nontrivial ultrafilter U on I . We refer to the Appendix for some
basic information on ultrafilters.
Let X be a Banach space. We denote by l , ( I , X ) the set of all the
bounded families ( x i ) i E ~
in X . The space l , ( I , X ) endowed with the
supremum norm
is a Banach space.
For each ( x i ) i EE~ l , ( I , X ) , (Ilxill)iEl is a bounded family of real numbers. Thus Theorem 28 implies that the limit 1imi-u IIxill exists.
It is not difficult to see that
N u ( X ) := {(xi) E &,(I, X )
:
lim llzill = 0)
Z-U
is a closed subspace of & ( I , X).
Definition 22. The ultrapower XU of X (with respect to the ultrafilter
U)is defined as the quotient space
48
We will denote by [xi] the element of
It is not difficult to check that
=
II[.illl
Xu associated to (xi) E &,(I, X).
p%Ibill.
Remark 23.
(1) The space X can be identified with the subspace of all the constant
classes in X u . Equivalently, the map
z
E
x
-
[ x , x , .. .] E xu
is an isometric embedding.
(2) The ultrapower ( X * ) u of X * can be identified with a subspace of
the dual ( X u ) *of X u by means of the map defined by
( [ f i l ,[xi]) := p+%fi(Zi),
where [ f i ] E ( X * ) u and [xi]E Xu.
In general ( X * ) uis strictly contained in (Xu)*.
See the Remarks
and Notes at the end of this section.
Example 8. (P.L.R. for ultrapowers)15 The ultrapower (X*),is a local
dual of Xu.
The following result is an extension of the P.L.R. for ultrapowers.
Theorem 24. l3 A subspace Z of X * is a local dual of X i f and only if
Zu is a local dual of Xu.
The are other characterizations of the subspaces of X* which are local
dual spaces of X in terms of ultrapowers.
Let Q : Zu -+ X* be the map defined by
Q[zi] :=weak*- lim zi E X*.
2 - 4
-
Theorem 25. A subspace Z c X* is a local dual of X if and only if there
Zu such that
exist a n ultrafilter U and a n isometric operator T : X *
QT = 1x8 and T ( z ) = [ z ,z , z , . ..], Vz E Z.
NOTES AND REMARKS
(1) Superreflexivity (see for its definition) is the local property that
corresponds to reflexivity. More precisely,
X superreflexive H Xu reflexive;
.3 ( X * ) u =
(Xu)*.
49
(2) There is a P.L.R. for ultrapowers of operators.
Every operator T : X + Y induces an operator TU : XU + YU
defined by Tu[zi]:= [Tzi].
It is easy to see that (Tu)*is an extension of (T*)u.Thus
N ( ( T * ) u )c N ( (Tu)*) = (Yu/R(Tu))*.
Example 9. Let T E B ( X ,Y ) . Then N ( (T*)u) is a local dual of
(YU/R(Tu))*.
lo that N ( (Tu)*)is f.d.r. in N ( (T*)u).
See the section of Notes and Remarks in Section 2 for the definition
0
of f.d.r. The proof of the present result is similar.
PROOF.It is proved in
9. Appendix: Ultrafilters
Here we will see a short introduction to the theory of ultrafilters which
could be useful to read the previous sections. For additional information
we refer to or l5.
The ultrafilters are a tool that allows us to get a limit for some families
of elements of a compact set. Each of these families have adherent points,
and the ultrafilter selects one adherent point for each family in such a way
that the linearity and some other properties of the limit are preserved.
Let I be a nonempty set. A filter 3 o n I is a non-empty family of
subsets of I satisfying
(1) A , B E 7 + A n B E 3.
(2) A E 3 , Ac B c I +B E
F.
Definition 26. An ultrafilter U o n I is a filter on I which is maximal with
respect to ordering by containment. That is,
3 filter on I and U c 3 implies 3 = U .
An ultrafilter on U on I is said to be nontrivial if it contains no finite
subsets of I .
We can define the convergence with respect to an ultrafilter as follows.
Definition 27. Let U be an ultrafilter on a set I and let K be a topological
space. We say that a family (ICi)iE1c K converges over U t o k E K if for
every neighborhood W of k ,
{ Z E I : kiEW} EU.
50
In this case we write Ic = 1imi-U
Ici.
The property of the ultrafilters which is the most useful for us is given
in the following result.
Theorem 28. Let K be a compact Hausdorff space and let U be an ultrafilter on I .
For each family ( k i ) i E l c K , the limit 1imi-u ki exists in K .
Finally let us give a detailed description of an application of Theorem
28 which appears several times in the paper.
Remark 29.
Let U be a nontrivial ultrafilter on the set Fi(X) of all finite dimensional
subspaces of X refining the filter associated t o the inclusion.
For each F E Fi(X), let LF be an operator from X into a dual Banach
space Y * such that
{IILFII : F
E
Fi(X)I
is bounded.
By Alaoglu’s Theorem, the closed unit ball of Y * is compact for the
weak*-topology. Therefore, for every x E X ,{LFx : F E Fi(X)} is contained in a weak*-compact set of Y* and we can define
Lx := weak*-
lim
E-iU,xE E
LFx.
Note that each 5 E X belongs eventually t o some subspace F .
In this way we define a map L : X --+ Y * ,and we can write
L =weak*- lim L F .
E-iU
References
1. A.G. Aksoy and M.A. Khamsi. Nonstandard methods in fixed point theory.
Universitext. Springer-Verlag,New York, 1990.
2. J. Bourgain. New classes of &,-spaces. Springer Lecture Notes in Math. 889,
1981.
3. P.G. Casazza and N.J. Kalton. Notes on approximation properties in separable
Banach spaces, London Math. SOC.Lecture Notes 158, Cambridge Univ. Press
(1990), 49-63.
4. S. Diaz. A local approach to functionals on L m ( p , X ) , Proc. Amer. Math. SOC.
128 (2000), 101-109.
5 . J. Diestel and J.J. Uhl, Jr. Vector measures. Math. Surveys 15. Amer. Math.
SOC.,Providence, 1977.
51
6. J. van Dulst. Reflexive and superreflexive Banach spaces. C.W.I. Math. Tracts
59. Amsterdam 1989.
7. G. Godefroy. Existence and uniqueness of isometric preduals: a survey, Amer.
Math. SOC.Contemporary Math. 85 (1989), 131-193.
8. G. Godefroy and N.J. Kalton. Approximating sequences and bidual projections,
Quarterly J. Math. Oxford 48 (1997), 179-202.
9. G. Godefroy, N.J. Kalton and P.D. Saphar. Unconditional ideals in Banach
spaces, Studia Math. 104 (1993), 13-59.
10. M. Gonzalez and A. Martinez-Abej6n. Local reflexivity of dual Banach spaces,
Pacific J. Math. 56 (1999), 263-278.
11. M. Gonzalez and A. Martinez-Abejbn. Local dual spaces of a Banach space,
Studia Math. 142 (2001), 155-168.
12. M. Gonzalez and A. Martinez-Abej6n. Local dual spaces of Banach spaces of
vector-valued functions, Proc. Amer. Math. SOC.130 (2002), 3255-3258.
13. M. Gonzalez and A. Martinez-Abej6n. Ultrapowers and subspaces of the dual
of a Banach space, Glasgow Math. J., to appear.
14. P. Harmand, D. Werner and W. Werner. M-ideals in Banach spaces and
Banach algebras. Lecture Notes in Math. 1547.Springer-Verlag, Berlin, 1993.
15. S. Heinrich. Ultraproducts in Banach space theory, J. Reine Angew. Math.
313 (1980), 72-104.
16. J. Johnson. Remarks o n Banach spaces of compact operators, J. Funct. Anal.
32 (1979), 304-311. dir
17. W.B. Johnson, H.P. Rosenthal and M. Zippin. O n bases, finite dimensional
decompositions and weaker structures in Banach spaces, Israel J. Math. 9
(1971), 488-506.
18. N.J. Kalton. Locally complemented subspaces and &-spaces f o r 0 < p < 1,
Math. Nachr. 115 (1984), 71-97.
19. A. Lima. T h e metric approximation property, norm-one projections and intersection properties of balls, Israel J. Math. 84 (1993), 451-475.
20. A. Martinez-Abej6n. An elementary proof of the principle of local reflexivity.
Proc. Amer. Math. SOC. 127 (1999), 1397-1398.
21. W. Rudin. Real and complex analysis, 3rd ed. McGraw-Hill, 1986.
22. B. Sims and D. Yost. Banach spaces with m a n y projections, Proc. Cent. Math.
Anal. Austral. Nat. Univ. 14 (1986), 335-342.
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53
ORBITS OF OPERATORS
VLADIMiR MULLER*
Institute of Mathematics,
Czech Academy of Sciences,
iitnli 25, 115 67 Praha 1, Czech Republic
E-mail: [email protected]
The aim of this paper is to give a survey of results and ideas concerning orbits
of operators and related notions of weak and polynomial orbits. These concepts
are closely related to the invariant subspace/subset problem. Most of the proofs
are not given in full details, we rather try to indicate the basic ideas. The central
problems in the field are also formulated.
Mathematics Subject Classification: primary 47A05, 47A15, 47A16, secondary 47A11.
Keywords: orbits, invariant subspace problem, hypercyclic vectors, weak
orbits, capacity, Scott Brown technique.
1. Introduction
Denote by B ( X ) the algebra of all bounded linear operators acting on a
complex Banach space X . Let T E B ( X ) . By an orbit of T we mean
a sequence {Tnx : n = 0,1, , ..} where x E X is a fixed vector.
The concept of orbits comes from the theory of dynamical systems. In
the context of operator theory the notion was first used by Rolewicz [30].
Orbits of operators are closely connected with the local spectral theory, the
theory of semigroups of operators [26], and especially, with the invariant
subspace problem, see e.g. [4].
The invariant subspace problem is the most important open problem of
operator theory. Recall that a subset M c X is invariant with respect to an
operator T E B ( X ) if T M c M . The set M is nontrivial if (0) # M # X.
*Supported by grant no. 201/03/0041 of GA CR.
54
Problem 1.1. (invariant subspace problem) Let T be an operator on a
Hilbert space H of dimension 2 2. Does there exist a nontrivial closed
subspace invariant with respect to T?
It is easy to see that the problem has sense only for separable infinitedimensional spaces. Indeed, if H is nonseparable and x E H any nonzero
vector, then the vectors x , T x ,T 2 x , .. . span a nontrivial closed subspace
invariant with respect to T .
If dim H < 00, then T has at least one eigenvalue and the corresponding
eigenvector generates an invariant subspace of dimension 1. Note that the
existence of eigenvalues is equivalent to the fundamental theorem of algebra
that each nonconstant complex polynomial has a root. Thus the invariant
subspace problem is nontrivial even for finite-dimensional spaces.
Examples of Banach space operators without nontrivial closed invariant
subspaces were given by Enflo [8],Beauzamy [3] and Read [28]. Read [29]
also gave an example of an operator T with a stronger property that T has
no nontrivial closed invariant subset.
It is not known whether such an operator exists on a Hilbert space. This
“invariant subset problem” may be easier than Problem 1.l.
Problem 1.2. (invariant subset problem) Let T be an operator on a
Hilbert space H . Does there exist a nontrivial closed subset invariant with
respect to T?
Both Problems 1.1 and 1.2 are also open for operators on reflexive Banach spaces. More generally, the following problem is open:
Problem 1.3. Let T be an operator on a Banach space X . Does T* have
a nontrivial closed invariant subset/subspace?
It is easy to see that an operator T E B ( X ) has no nontrivial closed
invariant subspace if and only if all orbits corresponding to nonzero vectors
span all the space X (i.e., each nonzero vector is cyclic).
Similarly, T E B(X)has no nontrivial closed invariant subset if and
only if all orbits corresponding to nonzero vectors are dense, i.e., all nonzero
vectors are hypercyclic.
Thus orbits provide the basic information about the structure of an
operator.
55
Typically, the behaviour of an orbit {Tnz : n = 0,1,. . . } depends much
on the initial vector x E X. An operator can have some orbits very regular
and other orbits extremely irregular.
Example 1.4. Let H be a separable Hilbert space with an orthonormal
basis {eo, e l , . . . }. Let S be the backward shift, i.e., S is defined by Seo = 0
and Sei = ei-l (i 2 1). Consider the operator T = 2 s . Then:
c H such that IIT"z1I -+ 0 (z E M I ) ;
(ii) there is a dense subset M2 c H such that llTnzll
00
(x E M z ) ;
(iii) there is a residual subset M3 c H such that the set { P z : n = 0 , 1 , . . . }
(i) there is a dense subset M I
4
is dense in H for all z E M3.
As the set MI it is possible to take the set of all finite linear combinations
of the basis vectors ei. Properties (ii) and (iii) follow from general results
that will be discussed in the subsequent sections.
The paper is organized as follows. In the following section we study
regular orbits. Of particular interest are the orbits satisfying I(Tnzll 4 00.
It is easy to see that if an operator T has such an orbit, then it has a
nontrivial closed invariant subset {Tnz : n = 0 , 1 , . . . }-.
In the third section we study the other extreme - hypercyclic vectors,
i.e., the vectors with very irregular orbits.
In the subsequent sections we study weak and polynomial orbits. A
weak orbit of T is a sequence {(Tnrc,rc*) : n = 0,1,. . . } and a polynomial
orbit of T is a set of the form {p(T)a: : p polynomial}, where z E X and
x* E
x*.
Polynomial orbits are closely related with the notions of capacity and
local capacity of an operator. These concepts are studied in Section 6.
In the last section we discuss the Scott Brown technique which is the
most efficient method of constructing invariant subspaces of operators. On
an illustrative example we show the basic ideas of the method, which are
closely connected with orbits.
For simplicity we consider only complex Banach spaces. However, some
results concerning orbits remain true also for real Banach spaces. In particular, all results based on the Baire category theorem remain unchanged
for real Banach spaces.
Although the invariant subspace problem is usually formulated for complex Hilbert spaces, the corresponding question for real spaces (of dimen-
56
sion 2 3) is also open; it is very easy to find an operator on a 2-dimensional
real Hilbert space without nontrivial invariant subspaces.
2. Regular orbits
Let X be a complex Banach space and let T E B ( X ) . If T is power bounded
(i.e., sup, llTnll < 00) then all orbits are bounded. The converse follows
from the Banach-Steinhaus theorem.
Theorem 2.1. Let T E B ( X ) . Then T is power bounded if and only if
sup, llTnxll < 00 for all x E X .
A more precise statement is given by the following theorem. Recall that
a subset M c X is called residual if its complement X \ M is of the first
category. Equivalently, M is residual if and only if it contains a dense G6
subset.
Theorem 2.2. Let T E B ( X ) and let (a,) be a sequence of positive numa, = 0 . Then the set of all points x E X with the
bers such that
property that llTnxll 2 a,llTnll f o r infinitely many n is residual.
Proof. The statement is trivial if T is nilpotent. In the following we
assume that T n # 0 for all n.
For k E N set
hfk =
x : there exists n 2 k such that IIT"x[I > a,llTn(l}.
is an open set. We prove that Mk is dense. Let x E x and
{x
E
Clearly Mk
> 0. Choose n 2 k such that
< 1. There exists z E X of norm one
such that llTnzll > u , E - ~ I I T ~
Then
~~.
E
+
and so either llTn(z EZ)II > a,lITnll or llT"(x - E Z ) ~ )> u,llTnll. Thus
either z EZ E Mk or z - E Z E Mk, and therefore dist {z, Mk} 5 E . Since
x and E were arbitrary, the set Mk is dense.
By the Baire category theorem, the intersection
k f k is a dense
G6 set, hence it is residual. Clearly each z E
Mk satisfies llT"x11 2
u,llTnll for infinitely many n.
0
Denote by r ( T ) = max{IXI : X E a(T)} the spectral radius of an
operator T E B ( X ) . By the spectral radius formula we have T ( T ) =
limn+co (ITn II l / n = inf, IITn(ll/n. Recall that r(Tn) = ( r ( T ) ) " for all
n.
+
57
For z E X let r,(T) denote the local spectral radius defined by r,(T) =
limsup, jo3 IIT"II1/" (the limit limn+o3 IIT"zlI1lnin general does not exist).
The local spectral radius plays an important role in the local spectral theory.
Note that the resolvent z H ( z - T)-l =
$& is analytic on the set
{ z : IzI > r ( T ) } .Similarly, the local resolvent z H (z-T)-lz = C:=,
can be analytically extended to the set { z : IzI > r,(T)}.
It is easy to see that r,(T) 5 r ( T )for all z E X.
Corollary 2.3. cf. [31] Let T E B ( X ) . Then the set {z E X : r,(T) =
r ( T ) } is residual.
c,"==,
Proof. Let a, = n-l. By Theorem 2.2, there is a residual subset M c X
such that for each z E M we have llTnzll 2 n-lIITnll for infinitely many
n. Thus
for all z E M .
0
As we have seen, it is relatively easy to construct vectors x such that
infinitely many powers Tnx are large. It is much more difficult t o construct
orbits such that all powers Tnz are large in the norm. The result (and
many other results concerning orbits) is based on the spectral theory and
therefore it is valid only for complex spaces. For real Banach spaces see
Remark 2.14.
Denote by ae(T)the essential spectrum of T E B ( X ) ,i.e., the spectrum
,
K ( X ) is the ideal of all
of p(T) in the Calkin algebra B ( X ) / K ( X ) where
compact operators on X and p : B ( X )
B ( X ) / K ( X )is the canonical
projection. Equivalently, a e ( T )= {A E C : T - X is not Fredholm}. Let
r e ( T )denote the essential spectral radius, r e ( T )= max{IXI : X E a e ( T ) } .
If X is an infinite dimensional Banach space and T E B ( X ) then a e ( T )
is a nonempty compact subset of a ( T ) . Moreover, the difference a(T) \
a,(T) is equal to the union of some bounded components of C \ ae(T)
and of at most countably many isolated points. In particular, if X E a ( T )
belongs to the unbounded component of C \ ae(T)then X is an isolated
point of the spectrum a ( T ) ,it is an eigenvalue of finite multiplicity and the
corresponding spectral subspace is finite dimensional.
Denote further by ane(T)the essential approximate point spectrum of
T , i.e., a,,(T) is the set of all complex numbers X such that
-
inf{ l
l ( -~ ~ ) z l:lz E M , IIzlI = I }
=
o
58
for every subspace M c X with codimM < 00.
It is easy to see that X $ a T e ( T )if and only if dimKer (T - A) < 00
and T - X has closed range, i.e., if T - A is upper semi-Fredholm. It is
known [14] that aTe(T)contains the topological boundary of the essential
spectrum a,(T). In particular, one(T)is a nonempty compact subset of
the complex plane for each operator T on an infinite dimensional Banach
space X.
The elements of the essential approximate point spectrum a,,(T) are
)
are "approxvery useful for the study of orbits. For each X E a T e ( Tthere
imate eigenvectors" - vectors x E X of norm 1 such that II(T - X)xll is
arbitrarily small. Moreover, the approximate eigenvectors can be chosen in
an arbitrary subspace of finite codimension. This property is particularly
useful in various inductive constructions.
The following result was proved in [20]; for Hilbert space operators see
141.
Theorem 2.4. Let T be an operator on a Banach space X , let E > 0 and
let (a,) be a sequence of positive numbers such that limn+m a, = 0. Then:
(i) there exists a vector z E X such that llxll
a, . r ( F ) for all n;
< sup, a,
+E
and llTnxll 2
(ii) there exists a dense subset of points x E X such that llTnxll 2 anr(Tn)
for all but a finite number of n.
Outline of the proof. Let X E a ( T ) satisfy
two cases:
1x1 = r ( T ) . We distinguish
(a) Suppose that r ( T ) > r e ( T ) .Then X is an eigenvalue of T . The corresponding eigenvector 2 of norm one satisfies ((T"lcll= IIX"z(( = r(T") for
all n.
Moreover, X is an isolated point of the spectrum of T and the spectral
subspace Xo corresponding to X is finite dimensional. Let u E X \ X O .
It is easy to verify that there is a positive constant c = c(u) such that
llTjull 2 c . r ( T j ) for all j. Thus in this case the set of all points satisfying
(ii) is even residual.
for all E > 0, n E N and A4 c X of
(b) Let r ( T ) = r e ( T ) .Since X E uTe(T),
finite codimension there exists x E M such that IIxlI = 1 and II(Tj -Xj)xlI <
E ( j 5 n). These "approximate eigenvectors" are basic building stones
used in the construction of the vector with the required properties.
We indicate the proof for Hilbert space operators.
59
Without loss of generality it is possible to assume that 1 > a1 > a2 >
r ( T ) = 1 and X = 1. For k 2 0 set r k = min{j : aj < 2-'}.
w e construct inductively vectors X k E
of norm 1 such that Tjxk =
xk ( j 5 rk) and
e
,
x
< k , j 5 Tk)
(note that the subspace { u : Tju I Tjxi (i < k , j 5 r k ) } is of finite
T'Xk 1@ X i
(2
codimension) .
2-2+lzi.
Set x =
Let rk-1 < j 5 7-k. Since Tjxi 1Tjxk
xFl
(i # k), we have
Thus x satisfies llTjxll 2 aj for all j.
The full statement of Theorem 2.4 can be obtained by a modification
of this argument; we omit the details.
0
For Banach spaces the proof is a little bit more complicated. The basic
idea is t o use instead of the orthogonal complement of a finite dimensional
subspace (which was in fact used here) the following lemma.
Lemma 2.5. Let X be a Banach space, let F c X be a finite dimensional
subspace and let E > 0. Then there exists a subspace M c X of finite
codimension such that
I b + f II 2 (1 - E)max{llflli
lle11P)
for all f E F and m E M .
An immediate consequence of Theorem 2.4 is the following corollary.
Corollary 2.6.Let T E B ( X ) . Then:
(i) the set { x E X : liminf,,,
JJT"xJJ1/"
= r ( T ) }is dense;
llTnxlllln = r ( T ) } is residual;
(ii) the set { x E X : limsup,,,
(iii) the set of all x E X such that the limit limn+, I(Tnxl('lnexists (and
is equal to r ( T ) )is dense.
As another corollary we get that the infimum and the supremum in the
spectral radius formula
60
can be exchanged.
Corollary 2.7. Let T E B(X).Then
Example 2.8. Let H be a separable Hilbert space with an orthonormal
basis ( e j : j = 0,1,. . . } and let S be the backward shift, Seo = 0, Sej =
ej-1
llS"zlll/" = 0} is
( j 2 1). Then the set {z E H : liminf,,,
residual.
Since r ( S ) = 1 and the set {z E H : limsup,,,
IIS"zlll/" = l} is also
residual, we see that the set {z E H : the limit limn+, IISnzII1/"exists}
is of the first category (but it is always dense by Corollary 2.6).
Proof. For k E N let
Mk
=
{ x E X : there exists n L k such that IlS"x1I < k-"}.
Clearly Mk is an open subset of X . Further, Mk is dense in X. To see
this, let u E X and c > 0. Let u = CEoajej and choose n 2 k such that
Cj"=
lql2
, < c2. Set y
= CyIi a j e j . Then IIy - uII < E and S"y = 0.
Thus y E Mk and Mk is a dense open subset of X.
By the Baire category theorem, the set A4 = nE"=,k is a dense G6
subset of X, hence it is residual.
Let x E M . For each k E N there is an n k 2 k such that IJSnkzll<
and so liminf,,,
IIS"zJJ'/n= 0.
0
It is also possible to combine conditions of Theorems 2.2 and 2.4 and to
obtain points z E X with llTnxl\ 2 a,. IITnII for all n; in this case, however,
there is a restriction on the sequence (a,).
Theorem 2.9. Let T E B ( X ) , let (a,) be a sequence of positive numbers
such that C,U;'~< 00. Then there exists x E X such that llTnxll 2
anlITnll for all n. There is a dense subset L c X such that for each x E L
there is a k E N with the property that
llTnxll 2 anllTnll
(n2 k).
Outline of the proof: Fix k E N. We indicate the construction of a
vector x satisfying llTjxll _> ajIITjll ( j 5 k). The vector satisfying this
relations for all n can be then obtained by a limit procedure.
61
Let k E N be fixed. For j = 1 , 2 , . . . ,k fix a vector z j E X of norm
one which almost attains the norm of T j , i.e., llTjxll IlTjII (we omit the
exact calculations).
L e t A = { X = ( X l , ...,X k ) E ~ ' : l X j 1 5 a ~ ' 3 f o r a l l j } . F o r X E A l e t
k
U A = Cj=l
X j z j . Consider the Lebesgue measure p on A.
For j = 1,.. . , k let Aj = {A E A : llTju~II< ajIITjll}. The basic idea
of the proof is to show that p ( A \ Us=l,Aj) > 0, which means that there
exists X E A such that llTju~II2 aj IITJ11 for all j = 1 , . . . ,k. For details
see [24].
A better estimate can be obtained if we replace the norm llSJJof an operator 5' E B(X) by the quantity IISIIp = inf{ llSlMll : M c X,codimM <
4.
If S is an operator on a separable Hilbert space H then IISJJ,coincides
with the essential norm llSlle = inf{llS KII : K E K ( H ) } .
+
For the proof of the next result see [24].
Theorem 2.10. Let T E B ( X ) . Let (a,) be a sequence of positive numbers
satisfying Cna, < co. Then there exists z E X such that JJTnzJJ
2
a,IITnlll, for all n.
The results of Theorems 2.9 and 2.10 can be improved for Hilbert space
operators, see [4].
Theorem 2.11. Let T be an operator on a Hilbert space H , let (a,) be a
sequence of positive numbers.
(i) if Enan < 00 then there exists z E H such that IJTnzll2 a,JJT"II for
all n E N;
(ii) if
a: < co then there exists z E H such that IIT"zll 2 a,((Tn((,for
all n E N.
C,
The following result is true for Hilbert space operators; in Banach spaces
it is false.
Theorem 2.12. [4] Let T be a non-nilpotent operator on a Hilbert space
= 0 0 } is residual.
H . Then the set {x E H : C
n
62
Example 2.13. Let X be the
space with the standard basis {ei :
i = 0 , l ...}. Let T E B(X) be defined by Teo = 0 and Ten =
- 2
< 00 for all z E X.
( n 2 1). Then C
( ,) e,-l
n
This can be verified by a direct calculation, see [24].
Remark 2.14. Some results from this section remain true for real Banach
spaces as well, see [24]. This is true for Theorem 2.2.
Theorem 2.4 can be reformulated as follows: if a, > 0, a, -+ 0, then
there exists a dense subset L c X such that for each
constant c > 0 with llTnzl(> ca,r(T)" for all n.
2
E L there is a
Theorem 2.9 can be modified in the following way: let T be an operator
on a real Banach space X, let (a,) be a sequence of positive numbers such
that C,a~" < 00. Then there exists z E X such that IITnz((2 a,(lT"((
for all n. There is a dense subset L c X such that for each z E L there is
a k E N with the property that
IIT"zll 2 anllTnll
( n 2 k).
3. Hypercyclic vectors
Vectors with extremely irregular orbits are called hypercyclic. More precisely, a vector z E X is called hypercyclic for an operator T E B ( X ) if the
set {Tnz : n = 0, 1, . . . } is dense in X. An operator T is called hypercyclic
if there is at least one vector hypercyclic for T .
Recall also that a vector z E X is called cyclic for T E B ( X ) if the set
{ p ( T ) z: p polynomial} is dense in X, and supercyclic for T if {AT"%: X E
c,n = 0 , l . . . }- =
These notions make sense only for separable Banach spaces. It is easy
to see that an operator in a non-separable Banach space can not have cyclic
(supercyclic, hypercyclic) vectors. Moreover, it is not difficult to show that
there are no hypercyclic operators on finite-dimensional Banach spaces (this
follows from the fact that T* has eigenvalues, cf. the proof of Theorem 3.2
below). In the rest of this section we assume that all Banach spaces are
infinite dimensional and separable.
It is easy to find an operator that is not hypercyclic. For example, any
contraction (or more generally, a power bounded operator) is not hypercyclic. On the other hand, if T has at least one hypercyclic vector then
almost all vectors are hypercyclic.
x.
63
Theorem 3.1. Let T E B ( X ) be a hypercyclic operator. Then there is
a residual set of vectors hypercyclic for T .
Proof. Let x E X be hypercyclic for T . For each Ic E N the vector T k x is
hypercyclic for T , and so the set of all hypercyclic vectors is dense.
Let ( U j ) be a countable base of open subsets in X . It is easy to see that
the set of all vectors hypercyclic for T is equal to
U, T - n U j , which is a
Gs subset.
0
nj
Theorem 3.2. Let T E B ( X ) be a hypercyclic operator. Then there
exists a dense linear manifold L c X such that each nonzero vector in L is
hypercyclic for T .
Proof. We show first that T* has no eigenvalues. Suppose on the contrary
that there are X E C and a nonzero vector x* E X * such that T*x*= Ax*.
Let x E X be a hypercyclic vector for T . Then
C = {(T,x,x*) : n = 0 , 1 ,
. . . } - = ( x , x * ) * ( X ~ : ~ = O , ,...
~ }-.
It is easy to see that the last set can not be dense in C. Thus X is not an
eigenvalue of T * ,and so (T - X)X is dense in X for each X E C.
Let x be a hypercyclic vector for T . We show that p(T)x is also hypercyclic for each nonzero polynomial p . Write p ( z ) = a ( z - XI) . . . (2 - A),
where a # 0, XI,. . . ,A, E C. Then
(T"p(T)x: n = 0,1,. . . } = a(T - X i ) . . . (T - X,){Tnx : 72 = 0 , 1 , . . . }.
The last set is dense in X since x is hypercyclic for T , and the operators
T - X j have dense ranges for each j. Thus p(T)x is hypercyclic for T . 0
The following criterion provides a simple way of constructing hypercyclic
vectors, see [16], [ l l ] . It also implies property (iii) in Example 1.4.
Theorem 3.3. Let T E B ( X ) . Suppose that there is an increasing sequence of positive integers (nk)such that:
(i) there is a dense subset X O c X such that lim~--tmTnkx
4 0 for all
xE
(ii)
T"kBx is dense in X , where Bx denotes the closed unit ball in X .
Then T is hypercyclic. By Theorem 3.1, this means that the set of all
hypercyclic vectors is residual.
x,;
Uk
Conversely, suppose that T is hypercyclic. Then it is not difficult to
show that T satisfies both conditions (i) and (ii), but not necessarily for
64
the same subsequence ( n k ) . Thus the conditions in Theorem 3.3 are close
to the notion of hypercyclicity (cf. Problem 3.12).
A similar criterion may be used to construct closed infinite dimensional
subspaces consisting of hypercyclic vectors, see [19], [12] and [17].
Theorem 3.4. Let T E B ( X ) . Suppose that T satisfies the conditions
of Theorem 3.3 and that the essential spectrum a,(T) intersects the closed
unit ball. Then there is a closed infinite dimensional subspace M c X such
that each nonzero vector in M is hypercyclic for T .
Theorems 3.1 - 3.4 indicate that hypercyclic vectors and operators are
quite common, that in some sense it is a typical behavior of an orbit.
Similarly as in Theorem 3.1 it is possible to show that the set of all
hypercyclic operators on a Banach space X is a Gg set. It is not dense
since the operators with IlTll < 1 can not be hypercyclic. Thus the set of
all hypercyclic operators is a residual subset of its closure.
By [15],it is possible to characterize the closure of hypercyclic operators
on a separable Hilbert space. For Banach spaces such a characterization is
not known.
Theorem 3.5. Let H be a separable Hilbert space, let T E B ( H ) . Then T
belongs to the closure of hypercyclic operators if and only if the following
conditions are satisfied:
(i) the set a w ( T )U { z E C : IzI = 1) is connected;
(ii) ao(T)= 0;
(iii) ind (A - 5") 2 0 for all X E C such that X - T is semi-Fredholm.
nKEK(H)
a(T+
Here a w ( T )denotes the Weyl spectrum of T , a w ( T )=
K ) . Equivalently, X $ aw ( T )if and only if T - X is Fredholm and ind (T A) = 0.
Recall that an operator S is called semi-Fredholm if it has closed range
and either dim ker S < co or codim SX < 00. The index of a semi-Fredholm
operator S is defined by ind S = dim ker S - codim SX.
Furthermore, ao(T) denotes the set of all isolated points of a ( T ) such
that the corresponding spectral subspace is finite dimensional.
Theorem 3.6. Let T E
for T . Then:
B(X)be an operator and let z E X be hypercyclic
65
(i) z is hypercyclic for T" for each n E N;
(ii) z is hypercyclic for AT for each X E C, 1x1 = 1;
(iii) if T is invertible then T-' is hypercyclic.
The first statement of Theorem 3.6 was proved in [2]. For (ii) see [MI.
The third statement follows from the observation that T is hypercyclic if
and only if for all nonempty open subsets U, V c X there exists n E N such
that T"U n V # 0.
Although it is relatively easy to construct an operator with a residual
set of hypercyclic vectors (see Example 1.4), it is extremely difficult to construct an operator with all nonzero vectors hypercyclic. The first example
of this type was constructed by Read [29] on the space e l . Equivalently,
such an operator has no nontrivial closed invariant subset. It is an open
problem whether this can happen in Hilbert spaces, cf. Problems 1.2 and
1.3.
The next result shows that such an operator must satisfy certain rather
narrow conditions on the norms IITn(l.
Theorem 3.7. Let T be an operator on a Banach space X which has no
nontrivial closed invariant subsets. Then r ( T ) = re(T)= 1, supn llT"ll =
00, C , ((Tnll-2/3
< 00 and C, llTn(lil< 00.
If X is a Hilbert space then
C,
JJTnJJ-'
< co and
C,
JJTn]1,1'2< 00.
Indeed, if T does not satisfy the conditions above, then either T is power
bounded or there exists a vector z E X with llTnzll + 00, see Theorems
2.4, 2.9, 2.10 and 2.11. Hence {Tnz : n = 0 , l . . , }- is a nontrivial closed
invariant subset with respect to T .
Thus it is a very interesting question for which operators there are orbits
satisfying llTnzll + co.
Problem 3.8. What are the best exponents in Theorem 3.7?
Example 3.9. There is an operator T on a Hilbert space H such that
llTn(l+ co and there is no z E H with \lTnxll + 00, see [4]. As an example
it is possible to take a unilateral weighted shift with suitable weights; the
operator satisfies ((T"1)= (lnn)1/2.
It is also possible to construct an operator T E B ( H ) such that
inf, IIT"zII = 0 and SUP, llTnzll = 00 for all nonzero vectors z E H .
66
We finish this section with some other open problems.
Problem 3.10.
Let T be a Hilbert space operator such that
limn+w llTnll = 00 and the norms llTnll form a nondecreasing sequence.
Does there exist a vector z E H such that IIT"zll --t oo?
Problem 3.11. Is the characterization of the closure of hypercyclic operators (Theorem 3.5) true also for Banach spaces?
Problem 3.12. Does there exist a hypercyclic operator T E B(X)that
does not satisfy conditions of Theorem 3.3?
There are other equivalent formulations of this problem. The most
interesting reformulation is: does there exist a hypercyclic operator T such
that T @ T is not hypercyclic, see [5]?
Problem 3.13. An operator T E B ( X ) is called weakly hypercyclic if
there exists a vector z E X such that the orbit {Tnz : n = 0,1,. . . } is
weakly dense in X, see [9]. Does there exists a weakly hypercyclic operator
that is not hypercyclic? Must a weakly dense orbit be norm dense?
Note that the corresponding notion of weakly cyclic vectors makes no
sense since a weakly closed linear manifold is automatically closed by the
Hahn-Banach theorem.
4. Weak orbits
Weak orbits were introduced and first studied by van Neerven [26]. Many
results for orbits of operators can be modified also for weak orbits. For a
survey of results see e.g. [26], [24].
The following three results are analogous t o the corresponding statements for orbits.
Theorem 4.1. Let T E B(X)and let (a,) be a sequence of positive
numbers such that limn--rwan = 0. Then the set of all pairs (x,z*)E
X x X* with the property that I(Tnz,z*)l > a,IITnll for infinitely many n
is a residual subset of X x X*.
Theorem 4.2. Let T be an operator on a Banach space X, let (a,) be
a sequence of positive numbers such that C ,
< 00. Then there exist
z E X and x* E X* such that I(Tnz,z*)l
2 anIITnll for all n.
Theorem 4.3. Let T E B(X).Then:
67
(i) the set {(z,z*) E X x
X* : liminf
I(Tnz,z*)(l/n
= r ( T ) }is dense;
n-im
(ii) the set { (x,z*) E X x
X* : limsup I(Tnx,x * ) ( l l n= T ( T ) }is residual;
n-+w
(iii) the set of all pairs
(5, z*)
E X x X*such that the limit lim
I(Tnx)I1/n
n-+m
exists (and is equal t o r ( T ) )is dense.
The statement analogous to Theorem 2.12 for weak orbits is not true:
Example 4.4. There exists an operator T on a Hilbert space H such that
< co for all x , y E H.
c
n
As an example it is possible to take the operator T = @ElSk, where
Sk is the shift operator on a (k + 1)-dimensional Hilbert space, see [24].
The statement analogous t o Theorem 2.4 for weak orbits is an open
problem:
Problem 4.5. Let T E B ( X ) , let (a,) be a sequence of positive numbers
satisfying lim,-,man = 0. Do there exist x E X and z* E X* such that
I(Tnx,z*))
2 a n r ( T n ) for all n?
The statement is false for real Banach spaces. A partial positive answer
is given in the following case which is important from the point of view of
the invariant subspace problem. Some other partial results were given in
P61.
Theorem 4.6. Let T be an operator on a Hilbert space H such that
1 E a ( T ) and Tnz -+ 0 for all z E H . Let (a,) be a sequence of positive
numbers satisfying limn+man = 0. Then there exists z E H such that
Re (T"x,5) > a, for all n.
Using Theorem 4.6 and techniques of [18] it is possible to obtain the
following result.
Theorem 4.7. Let T be a power bounded operator o n a Hilbert space H
satisfying r ( T ) = 1. Then there is a nonzero vector x E H such that x
is not supercyclic. Moreover, T has a nontrivial closed invariant positive
cone, i.e., there is a nontrivial closed subset M c H such that T M c M ,
M + M c M and t M c M (t 2 0 ) .
68
It is a natural question whether the previous result can be improved in
order to obtain an invariant real subspace.
Problem 4.8. Let T be a power bounded operator on a Hilbert space such
that r ( T ) = 1. Does T have a nontrivial closed invariant real subspace, i.e.,
does there exists a nontrivial closed subset M c H such that T M C M ,
M M c M and t M c M (t E R) ?
+
Problem 4.9. Is Theorem 4.7 true f o r operators on reflexive Banach
spaces ?
5. Polynomial orbits
If 2 is an eigenvector of T , T x = Xz for some complex A, then p(T)a: =
p(X)x for every polynomial p , and so we have a complete information about
the polynomial orbit {p(T)z: p polynomial}. Unfortunately, operators on
infinite dimensional Banach spaces have usually no eigenvalues. The proper
tool appears to be the notion of the essential approximate point spectrum
a?re(T).
The following result is an analogue of Theorem 2.4.
Theorem 5.1. [22] Let T be a n operator o n a Banach space X , let X E
axe(T).Let (a,) be a sequence of positive numbers with
a, = 0.
Then:
(i) there exists x E X such that
llP(T)X11 2 a d e g p IP(X>l
f o r every polynomial p ;
(ii) let u E X , E > 0. Then there exists y
C = C(E)such that IIy - uII 5 E and
IlP(T)Yll 2 C ' a d e g p '
E
X and a positive constant
b(X)l
f o r every polynomial p .
In the previous theorem we expressed the estimate of llp(T)x11 by means
of Ip(X)l where X was a fixed element of axe(T).Next we are looking for
Since da,(T) 3 aTe(T),
an estimate in terms of max{lp(X)( : X E aTe(T)}.
by the spectral mapping theorem for the essential spectrum ae we have
69
An important tool for the results in this section is the following classical
lemma of Fekete [lo]. It enables to estimate the maximum of a polynomial
on a (in general very complicated) compact set u&?) by means of its
values at finitely many points.
Lemma 5.2. Let K be a non-empty compact subset of the complex plane
and let k 2 1. Then there exist points U O ,ul,. . . ,U k E K such that
m={Ip(z)I : E K } I
(k
+ 1). Om=
IP(Ui)l
liSk
for every polynomial p with degp 5 k. Moreover, we have
k.
for all polynomials p with degp I
By using the previous lemma we can get [21], [23]
Theorem 5.3. Let T be an operator on a Banach space X, let
k 2 1. Then:
(i) if carda,,(T) 2 k
E
2 0 and
+ 1 then there exists x E X with ((Ic((= 1 and
for every polynomial p with degp I
k.
(ii) let x E X and E > 0. Then there exists
C = C(E)such that IIy - 5 E and
XI(
IMT)YII2 c
y E
X and a positive constant
(1 + degp)-(l+E) re(p(T))
for every polynomial p .
The proof of Theorem 5.3 is much simpler for operators on Hilbert
spaces. The same result for Banach space operators can be obtained by a
Dvoretzky’s theorem type argument. A simpler proof based on Lemma 2.5
is available for weaker estimates Ilp(T)xII 2 z(k;t)2re(p(T)) and llp(T)z/(_>
C . (1 degp)-(2+E)r,(p(T)),respectively.
The estimates in Theorem 5.3 (i) are the best possible.
+
Example 5.4.[23] Let k E N. There exists a Banach space X and an
operator T E B ( X ) such that for each x E X of norm one there is a
polynomial p of degree 5 k with llp(T)xll I
(k l)-lr,(p(T)).
+
70
6. Capacity
The notion of capacity of an operator (or more generally, of a Banach
algebra element) was introduced and studied by Halmos [13]. If T E B ( X )
then
capT = lim (capkT)llk= inf(capkT)l/k,
k
k-+m
where
capkT = inf { Ilp(T))): p E
PL}
and P; is the set of all monic (i.e., with leading coefficient equal to 1)
polynomials of degree k.
This is a generalization of the classical notion of capacity (sometimes
also called Tshebyshev constant) of a nonempty compact subset K of the
complex plane:
cap K = lim (capkK)llk= inf(capkK)l/k
k-cc
k
where
capkK = inf { l l p l l ~: p E P l }
and
llpl)=
~ sup{lp(z)l : z E K}.
The classical capacity cap K is equal to the capacity of the identical
function f ( z ) = z considered as an element of the Banach algebra of all
continuous functions on K with the sup-norm.
Another connection between these two notions is given by the following
main result of [13].
Theorem 6.1. capT = capa(T) for each operator T E B ( X ) .
Let x E X. The local capacity of T at x can be defined analogously.
We define
capk(T,x) = inf { IIp(T)xll p E P i }
and
cap (T,x) = lim sup capk(T,x)l/k
k+cc
(in general the limit does not exist).
It is easy to see that cap(T, x) 5 cap T for every x E X .
Note that there is an analogy between the spectral radius and the capacity of an operator:
71
r,(T) = limsup IIT~x))'~',
k-+m
capT = lim (cap kT)llk= inf(cap kT)'lk,
k+m
cap (T,x) = lim sup (cap k(T,z)) I l k .
k-+m
Furthermore, capT 5 r(T) and cap(T,x)
Theorem 6.2. Let T E B ( X ) . Then:
[a)
the set (x E
x : liminfk,,
I r,(T)
for all z E X.
capk(T,z)l/k= capT} is dense an
x;
(ii) the set {x E X : cap(T, x) = cap T} is residual in X;
(iii) the set {x
E
x : limk,,
cap k(T,x)llk = cap T} as dense in
x.
Outline of the proof. By Theorem 5.3, there is a dense subset of vectors
X with the property that IIp(T)zll 2 C * (1 degp)-2re(p(T)) for all
polynomials p . Thus we have
+
II: E
caPk(T,z)
inf{llP(T)zll : P E
pi}
2 C . (1+ k)-2inf{re(p(T)) : p E P i } = C(1+ k)-2capkae(T).
Hence
liminf(cap k ~ ) l / k 2 liminf(cap k a , ( ~ ) ) l /=~cap ae(T).
k-cm
k-m
Using the general relations between o(T) and a,(T), it is possible to see that
capo,(T) = capa(T). Hence, by Theorem 6.1,liminfk,,(cap
(T,z))'lk=
capT for all x in a dense subset of X.
The second statement requires a more refined arguments, see [24]. 0
An operator T E B ( X ) is called quasialgebraic if and only if cap T = 0.
Similarly, T is called locally quasialgebraic if cap ( T ,II:)
= 0 for every z E X.
It follows from Theorem 6.2 that these two notions are equivalent.
Corollary 6.3. An operator is quasialgebraic if and only if at is locally
quasialgebraic.
Corollary 6.3 is an analogy to the well-known result of Kaplansky: an
operator is algebraic (i.e. p ( T ) = 0 for some non-zero polynomial p ) if and
only if it is locally algebraic (i-e.,for every II: E X there exists a polynomial
p , # 0 such that p,(T)a: = 0).
7. Scott Brown technique
The Scott Brown technique is an efficient way of constructing invariant subspaces. It was first used for subnormal operators but later it was adapted
72
to contractions on Hilbert spaces and, more generally, to polynomially
bounded operators on Banach spaces. Some results are also known for
n-tuples of commuting operators.
The basic idea of the Scott Brown technique is to construct a weak
orbit {(T"x,
x*): n = 0, 1,. . . } which behaves in a precise way. Typically,
vectors x E X and x* E X* are constructed such that
(T"x,x*)=
{
0
1
nZ1;
n=0.
Equivalently,
(1)
(P(T)X,x*) = P ( 0 )
for all polynomials p . Then T has a nontrivial closed invariant subspace.
Indeed, either T x = 0 (and x generates a 1-dimensional invariant subspace)
or the vectors {Tnz : n 2 1) generate a nontrivial closed invariant subspace.
The vectors x and x* satisfying the above described conditions are constructed as limits of sequences that satisfy (1) approximately.
Let D = { z E CC : JzI < 1) denote the open unit disc in the complex
plane and T = { z E CC : IzI = 1) the unit circle. Denote by P the normed
space of all polynomials with the norm IJpJJ
= sup{)p(z)): z E D}. Let P*
be its dual with the usual dual norm.
Let $ E P*. By the Hahn-Banach theorem, $ can be extended without
changing the norm to a functional on the space of all continuous function on
T with the sup-norm. By the Resz theorem, there exists a Bore1 measure p
on T such that ((pL((
= 11$11 and $ ( p ) = J p d p for all polynomials p . Clearly,
the measure is not unique.
Let L1 be the Banach space of all complex integrable functions on T
with the norm llflll = (27r)-'J:",
If(eit))dt.
Of particular interest are the following functionals on P:
(i) Let X E D. Denote by Ex the evaluation at the point A, i.e., Ex is defined
by E x b ) = P ( 4
( P E PI.
(ii) Let f E L1. Denote by M f E
~ f ( p=) (27r)-1/"
P*the functional defined by
p(e"">(eit)dt
(pE
P).
--x
Then IlMfll F Ilflll.
The evaluation functionals Ex are also of this type. Indeed, for X E D
we have Ex = M p x , where Px(eit) = 5- 1-JX12 is the Poisson kernel. In
73
particular, if g = 1 then Mg(p)= p ( 0 ) for all p , and so Mg is the evaluation
at the origin.
(iii) Let k > 0 and let T : X -+ X a polynomially bounded operator with
constant k, i.e., T satisfies the condition Ilp(T)II 5 IclJpJJ
for all polynomials
p. Fix x E X and x* E X * . Let x @ x* E P* be the functional defined by
.(
@ X*>(P) =
(p(T)x,x*)
(P E P).
Since T is polynomially bounded, x @ x* is a bounded functional and we
have llx @x*II5 kllzll . llz*II.
Of course the definition of x@x* depends on the operator T but since we
are going to consider only one operator T , this can not lead to a confusion.
By the von Neumann inequality, any contraction on a Hilbert space is
polynomially bounded with constant 1. More generally, every operator on
a Hilbert space which is similar to a contraction is polynomially bounded.
Recall that there are polynomially bounded Hilbert space operators that are
not similar to a contraction. This was shown recently by Pisier [27] who
gave thus a negative answer to a well-known longstanding open problem
given by Halmos.
Denote by L" the space of all bounded measurable functions on with
the sup-norm. Since P c Loo = (L1)*,the space P inherits the w*-topology
from L".
Of particular importance for the Scott Brown technique are those functionals on P that are w*-continuous, i.e., that are continuous functions
w*) to @. Equivalently, these functionals can be represented by
from (P,
absolutely continuous measures.
The next result summarizes the basic facts about w*-continuous functionals on P .
Theorem 7.1.
(i) Let ( p n ) C P be a sequence of polynomials. Then p , s O if and only i f
(pn) is a Montel sequence, i.e., supn JJp,)I< 00 and p n ( z ) + 0 ( z E ID);
(ii) The w* closure of P in L" is the Hardy space H" of all bounded
functions analytic o n JD.
(iii) 1c, E P* is w*-continuous if and only i f it can be represented by a n
absolutely continuous measure. By the F. and M. Riesz theorem, in this
case each measure representing 11, is absolutely continuous. By the RadonNilcodym theorem, there exists f E L1 such that llflll = II11,II and 11, = M f .
74
(iv) Let @ E P* be w*-continuous. Let A c D be a dominant subset,
i.e., supxe,, If ( X ) l = 11 f l l for all f E H". Let E > 0. Then there are
numbers XI,. . . ,A, E A and 0 1 , . . . , a , E C such that CZ1jail 5 ll@ll and
[I@ - c:=, 11 < E .
&xi
Let T E B ( X ) be a polynomially bounded operator such that IIT"uII -+
0 for all u E X . Then all the functionals x @ x* can be represented by
absolutely continuous measures. Equivalently, these functionals are w*continuous, i.e., they are continuous on the space (P,w * ) . These results
can be shown using classical results from measure theory.
We summarize the results in the following theorem. Conditions (iii) and
(iv) are not necessary for our purpose, we include them only for the sake
of completeness.
Theorem 7.2. Let T be a polynomially bounded operator o n a Banach
space X . Suppose that IIT"uII .+ 0 f o r all u E X . Then:
(a) x @ x* can be represented by an absolutely continuous measure for all
x E X and x* E X * . Equivalently, x @ x* is w*-continuous;
(ii) the set {p(T)x: p E P , llpll 5 1) is precompact f o r all x E X ;
(iii) the functional p H (p(T)x,x*) extends to the w*-closure of P in Loo,
i.e., to the Hardy space H" of all bounded functions analytic o n D;
(iv) it i s possible to define the HOO-functional calculus, i.e., an algebraic
homomorphism @ : H" --+ B ( X ) such that Q(1) = I and @ ( z ) = T .
Moreover, @ i s ( w * , S O T ) continuous, i.e., the mapping h -+ @ ( h ) xis a
continuous function from ( H m ,w * ) to X for each x E X .
Now we are able to give an illustrative example how the Scott Brown
technique works.
Theorem 7.3. Let T be a contraction o n a Hilbert space H such that
a(T)n D i s dominant in D and IIT"xII --+ 0 f o r all x E H . Then T has a
nontrivial closed invariant subspace.
Outline of the proof. Without loss of generality we may assume that
neither T nor T* has eigenvalues. In particular, a,,(T) = a ( T ) .
The first step in the proof is that we can approximate (with an arbitrary
precision) the evaluation functionals Ex for X E ane(T)
by the functionals
of the type x @ x with x E H .
75
(a) Let X E ore(T),E
> 0, let z E H ,
IIzlI = 1 and II(T - X)zll
< E . Then
Indeed, we have
2ke
I ( p ( T ) z4, - P(X)I = I (q(T)(T- X)z, 4 1 I Il4J(T)II II (T - X > 4 l I q.
*
For the approximation procedure we need a stronger version of (a).
. . ,u, E H be given. Let X E a&!') and E
(b) Let u1,.
exists z E H of norm 1 such that z I( 2 1 1 , . . . , u,} and
1
1
5 €3
> 0. Then there
z - 8x11 < E ,
112€3Uill < &
llut €3 511
<&
( i = 1 , ..., n ) ,
(i = 1,.. . ,n).
Indeed, since X E orre(T),
we can choose a vector 5 I{ u ~ ,. .. ,u,} such
that II(T - X)zll is small enough. Thus the inequality llz 18z - Ex11 < E
follows from (a). Using the same estimates it is possible to obtain also that
JJz€3uzJJ
<&
( i = l , ..., n).
For the last inequality (note that the second and third inequalities are
not symmetrical!) it is possible to use the compactness of the set {p(T)ui:
llpll I 1,i = 1,.. . , T I } - , see Theorem 7.2 (ii). Indeed, it is possible to choose
z "almost orthogonal" to all vectors of the form p(T)ui where ( ( p ( I
( 1 and
2 = 1, ...,n.
In the following we use the fact that any w*-continuous functional can be
approximated by convex linear combinations of the evaluations at points of
oae(T),see Theorem 7.1 (iv). We show that if 1c, E P* is any w*-continuous
functional and z €3 y its approximation, then it is possible to find a better
approximation X I €3 y' of 1c, that is not too far from z €3 y.
(c) Let 1c, E P* be a w*-continuous functional, let z,y E H and
Then there are X I , y' E H such that
IIzI €3 YI - $11 < E ,
1Izl - 211 5 112 €3 Y - 1c,l11/2,
llYl - Yll 5 llz €3 Y -
E
> 0.
76
Indeed, by Theorem 7.1 (iv) there are elements X i , . . . ,An E a,,(T) and
nonzero complex numbers al, . . . ,an such that CZl Iai(I 112 8 Y - $11
and
Let 6 be a sufficiently small positive number.
By (b), we can find inductively mutually orthogonal unit vectors
ul,..
. , un E H such that
< 6,
IlUi 8 YII < 6,
1
1
5 €3 Uill
11~i€3ujII< b
(Iui€3 ui - Exi 11
+
(ifj),
< 6.
+
Set x’ = x CZ1*ui
and y’ = y CZ1Jai11/2ui. Since the vectors
u1,.
. . ,u,are orthonormal, we have llx’ - x1I2 =
(ail 5 llx €3 y - $11,
and similarly, lly’ - y1I2 5 11% 8 y - $11. Furthermore,
cy=l
n
n
n
n
provided 6 is sufficiently small.
(d) There are x , y E H such that x €3 y = €0. As it was shown above, this
implies that T has a nontrivial invariant subspace.
Set xo = 0 = yo. Using (c) it is possible to construct inductively vectors
( j E N)such that
xj,y j E H
IIxcj €3 yj - €011 I 2-2j,
Ilzj+l - xjll 5 llxj €3 yj - €
IlYj+l - Yjll I 2 - j .
Clearly the sequences
(xj)and
~llI
~ /2-i~
and
(yj) are Cauchy. Let x and y be their limits.
It is easy to verify that x €3 y = €0.
0
77
The condition that Tnx+ 0 for all x E
reduction argument.
H can be omitted by a standard
Theorem 7.4. [6]LetT be a contraction on a Hilbert space H such that
the spectrum a(T)nD is dominant in D.Then T has a nontrivial invariant
subspace.
Outline of the proof. Let M I = {x E H : Tnx 0 ) . It is easy to see
that M I is a closed subspace of H invariant with respect t o T.If M I = H
that T has a nontrivial invariant subspace by Theorem 7.3. Thus we can
assume without loss of generality that M I = ( 0 ) .
Since a subspace M c H is an invariant subspace for T if and only if M I
is an invariant subspace for T * ,we can do all the previous considerations
also for T* instead of T. Thus we can also assume that M2 = {x E H :
T * n ~0) = (0).
Contractions T E B ( H ) satisfying M I = ( 0 ) = Mz are called contractions of class C11 in [25]. It is proved there that such T is quasisimilar to
a unitary operator (i.e., there are a Hilbert space K , a unitary operator
U E B(K) and injective operators A : H t K, B : K
H with dense
ranges such that UA = AT and B U = TB).Consequently, T has many
0
invariant subspaces, see [25].
-+
-+
Theorem 7.4 is a classical application of the Scott Brown technique. By
refined methods it is possible to obtain the following much deeper result.
Theorem 7.5. [7] Let T be a contraction o n a Hilbert space H such that
a ( T ) contains the unit circle { z E C : Izl = 1). Then T has a nontrivial
closed invariant subspace.
Theorem 7.5 can be also generalized to the Banach space setting.
Theorem 7.6. [l]Let T be a polynomially bounded operator on a Banach
space X such that a(T)contains the unit circle. Then T* has a nontrivial
invariant subspace. I n particular, if X is reflexive then T has a n invariant
subspace.
Note that Theorem 7.6 is stronger than Theorem 7.5 even for Hilbert
space operators.
78
References
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subspaces II., to appear.
2. S.I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995),
374-383.
3. B. Beauzamy, Un ope‘rateur sans sous-espace invariant: simplification
de l’example de P. EnAo, Integral Equations Operator Theory 8 (1985),
314-384.
4. B. Beauzamy, Introduction to operator theory and invariant subspaces,
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(1999), 94-112.
6. S. Brown, B. Chevreau, C. Pearcy, Contractions with rich spectrum have
invariant subspaces, J. Operator Theory, 1 (1979), 123-136.
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to appear.
10. M. Fekete, Uber die Verteilung der Wurzeln bei gewissen algebraischen
Gleichungen mit ganzzahligen Koefficienten, Math. Z. 17 (1923), 228-249.
11. G. Godefroy, J.H. Shapiro, Operators with dense, invariant, cyclic vector
manifolds, J. Funct. Anal. 98 (1991), 229-269.
12. M. Gonzalez, F. Lebn, A. Montes, Semi-Fkedholm theory: hypercyclic
and supercyclic subspaces, Proc. London Math. SOC.81 (2000), 169-189.
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(1971), 855-863.
14. R. Harte and T. Wickstead, Upper and lower Fkedholm spectra IT.,Math.
Z. 154 (1977), 253-256.
15. D.A. Herrero, Limits of hypercyclic operators, J. Funct. Anal. 99 (1991),
179-190.
16. C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto, Toronto, 1982.
17. F. Le6n, A. Montes, Spectral theory and hypercyclic subspaces, Trans.
Amer. Math. SOC.353 (2000), 247-267.
18. F. Lebn, V. Miiller, Rotations of hypercyclic operators, to appear.
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(1996), 419-436.
20. V. Muller, Local spectral radius formula for operators on Banach spaces,
Czechoslovak J. Math. 38 (1988), 726-729.
21. V. Muller, Local behaviour of the polynomial calculus of operators, J.
Reine Angew. Math. 430 (1992), 61-68.
22. V. Muller, On the essential approximate point spectrum of operators, Int.
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Equations Operator Theory 15 (1992), 1033-1041.
23. V. Muller, Dvoretzky’s type result for operators on Banach spaces, Acta
Sci. Math. (Szeged) 66 (2000), 697-709.
24. V. Muller, Orbits, weak orbits and local capacity of operators, Integral
Equations Operator Theory 41 (2001), 230-253.
25. B. Sz.-Nagy, C. Foiq, Harmonic analysis of operators on Hilbert space,
North-Holland, Amsterdam, 1970.
26. J.M.A.N. van Neerven, The asymptotic behaviour of semigroups of operators, Operator Theory: Advances and Applications, vol. 88, Birkhauser,
Basel, 1996.
27. G. Pisier, A polynomially bounded operator on Hilbert space which is
10 (1997), 351-369.
not similar to a contraction, J. Amer. Math. SOC.
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SOC.16 (1984), 337-401.
29. C.J. Read, The invariant subspace problem for a class of Banach spaces
11. Hypercyclic operators, Israel J. Math. 63 (1988), 1-40.
30. S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22.
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Czechoslovak Math. J. 25 (1973), 483-492.
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81
GENERICITY IN NONEXPANSIVE MAPPING THEORY
EVA MATOUSKOVA*
Institut fiir Mathematik
Johannes Kepler Universitat
A-4040 Linz, Austria
Email: [email protected]. at
SIMEON REICH+
Department of Mathematics
The Technion-Israel Institute of Technology
32000 Haifa, Israel
Email: [email protected]
ALEXANDER J. ZASLAVSKI
Department of Mathematics
The Technion-Israel Institute of Technology
32000 Haifa, Israel
[email protected]. ac.il
We review the concepts of Baire’s categories, porosity, and null sets in metric and
Banach spaces, and then illustrate the generic approach to nonlinear problems by
presenting and discussing several simple examples of its applications to nonexpansive mapping theory. An extension theorem for contractive mappings is also
included.
Mathematics Subject Classification: Primary: 47H09; Secondary: 47H10,
54C20, 54350, 54352.
Keywords: Baire’s categories, contractive mapping, extension of mappings,
generic property, nonexpansive mapping, null set, porous set, strict contraction, well-posedness.
*Supported by Grant No. FWF-Pl6674-Nl2.
?Supported by the Israel Science Foundation founded by the Israel Academy of Sciences
and Humanities, Grant 592100.
82
1. Baire’s categories
Let X be a complete metric space. According t o Baire’s classical theorem,
the intersection of every countable collection of open, dense subsets of X
is dense in X . This rather simple, yet powerful result has found many
applications. In particular, given a property which elements of X may
have, it is of interest to determine whether this property is generic. In other
words, whether the set of elements which do enjoy this property contains
a residual subset, that is, a countable intersection of open, dense sets. A
generic property may be thought of as being, in some sense, “typical”. Such
an approach, when a certain property is investigated for the whole space X
and not just for a single point in X , has already been successfully applied
in many areas of Analysis (see, for example, [8, 9 , 27, 29, 301, Chapter 16
in [15],and the references mentioned there).
Recall that a set E c X is called nowhere dense if its closure contains
no nonempty open subset of X . Any countable union of nowhere dense sets
is said t o be of the first Baire category; all other subsets of X are of the
second Baire category. Thus Baire’s theorem can be rephrased as follows:
no complete metric space is of the first category.
After reviewing the related concepts of porosity and null sets in the
next two sections, we intend t o illustrate the generic approach t o nonlinear
problems by presenting and discussing several simple examples of its applications t o nonexpansive mapping theory. These applications involve the
convergence of iterates, contractive mappings, well-posedness, and strict
contractions. An extension theorem for contractive mappings, which may
be of independent interest, is also included.
2. Porous sets
In this section we discuss the concept of porosity [ 2 ] ,[7]-[9],
[27]-[30]
which
is a refinement of the notion of Baire’s first category.
Let (Y,
d ) be a complete metric space. We denote by Bd(y, r ) the closed
ball of center y E Y and radius r > 0. We say that a subset E c Y is porous
in (Y,d ) if there exist Q E ( 0 , l ) and rg > 0 such that for each r E (0, rg]
and each y E Y , there exists z E Y for which
& ( z , Qr) c &(Y, r ) \ E.
A subset of the space Y is called 0-porous in (Y,d ) if it is a countable union
of porous subsets in (Y,d).
Other notions of porosity have been used in the literature [2, 341. We
83
use the rather strong notion which appears, for instance, in [7]-[9],
[27],
[28],and which turns out to be appropriate for our purposes.
Since porous sets are nowhere dense, all a-porous sets are of the first
Baire category. If Y is a finite-dimensional Euclidean space R", then the
Lebesgue density theorem implies that a-porous sets are of Lebesgue measure 0. The existence of a non-a-porous set P c R" which is of the first
Baire category and of Lebesgue measure 0 was established in [33,341. It
is easy to see that for any a-porous set A c R", the set A IJ P c R"
also belongs t o the family E of all the non-a-porous subsets of R" which
are of the first Baire category and have Lebesgue measure 0. Moreover, if
Q E E is a countable union of sets Qi c Rn, i = 1,2,.. . , then there is a
natural number j for which the set Q j is non-a-porous. Evidently, this set
Qj also belongs to 1. Therefore one sees that the family E is quite large.
Also, every complete metric space without isolated points contains a closed
nowhere dense set which is not a-porous [35].
To point out the difference between porous and nowhere dense sets, note
that if E c Y is nowhere dense, y E Y and r > 0, then there are a point
a E Y and a number s > 0 such that B d ( a , s) c Bd(y, r ) \ E. If, however,
E is also porous, then for small enough r we can choose s = ar, where
a E ( 0 , l ) is a constant which depends only on E.
In general, when we look for either genericity or porosity results we are
faced with the problem of choosing an appropriate metric with the goal of
obtaining a large set of "good" elements. Usually, a given (function) space
can be equipped with several natural metrics and an optimal choice is not
possible. For example, in a genericity result we want the set of "good"
elements to be dense with respect t o a strong topology and to be a Gg set
with respect t o a weak topology. To overcome this difficulty, we used a
two-topology approach in [25]. In the setting of Banach space geometry
such an approach was first used in [ll]. In the context of porosity we need
to use a two-metric approach.
The concept of porosity with respect t o a pair of metrics was introduced
in [36]and used in [29,301. To define this concept, assume that Y is a
nonempty set and that d l , d2 : Y x Y + [ O , o o ) are two metrics which
satisfy d l ( z , y ) 5 dz(z,y) for all 5,y E Y .
We say that a subset E c Y is porous in Y with respect to the pair
(d1,dz) (or just porous in Y if the pair of metrics is understood) if there
exist (Y E (0,l)and T O > 0 such that for each r E (0, TO] and each y E Y ,
there exists z E Y for which & ( z , y) 5 r and B d l ( a , ar) n E = 8.
A subset of the space Y is called a-porous in Y with respect to ( d l , dz)
84
(or just a-porous in Y if the pair of metrics is understood) if it is a countable
union of porous (with respect to ( d l ,d 2 ) ) subsets of Y .
Note that if d l = d2, then by Proposition 1.1 of [36] our definitions
reduce to those in [7]-[9],[27], [28].
Notice also that the porosity of a set with respect to one of the metrics dl
or d2 does not imply its porosity with respect to the other metric. However,
it is shown in [36], Proposition 1.2, that if a subset E c Y is porous with
respect to ( d l , d 2 ) , then E is porous with respect t o any metric d which is
between d l and d2, that is, c l d l 5 d 5 c2d2 for some positive c1 and c2.
In the next section we discuss the concept of null sets in infinitedimensional Banach spaces. In such spaces the relationships between aporous sets and null sets are not yet fully understood. For instance, in [20]
the authors construct a subset M of the separable Hilbert space l 2 which
is porous in a very strong sense (arbitrarily close to each z E M , there
is a “hole”, that is, a ball of radius one, which is completely outside M ) ,
but at the same time very large in the sense of measure. This construction
was generalized in [21] to separable superreflexive Banach spaces. Such
phenomena are impossible in a finite-dimensional setting.
3. Null sets
In the previous two sections we reviewed two related useful methods for finding points which are plentiful in either a topological or a stronger, metric
sense. This approach to proving that a certain subset is dense is, however,
not always suitable. For instance, let f : R” -+ R be a Lipschitz function.
Then, by a theorem of Rademacher, f is differentiable almost everywhere.
Nevertheless, the set of points where f is differentiable can be of the first
category. This raises the question of how we are going to prove a generalization of Rademacher’s theorem to infinitely many dimensions, as, obviously,
Baire’s theorem is not going to be of help. A natural idea, which indeed
turns out to work, is to introduce a concept analogous to Lebesgue null
sets, and hence also a notion of a property holding almost everywhere, in
separable infinite-dimensional Banach spaces.
Here are some natural requirements such a family of null sets should
satisfy:
(i) Every translate of a null set is null. This is necessary for the application we have in mind. Let a Lipschitz function f be defined on a
set A and let B be a translate of A. Define a new Lipschitz function
g on B by composing f with an appropriate translation. Then the
85
set of points where g is not differentiable is just a translate of the
set where f was not differentiable.
(ii) Nonempty open sets are not null. This is natural, as we want to
use null sets to prove that certain subsets of their complements are
dense.
(iii) A countable union of null sets is null. This will be a useful tool.
Often we can describe the desired property by excluding countably
many “bad” cases. If each of them happens only on a null set, then
their union is also a null set, and hence the desired property takes
place almost everywhere.
All of the above properties are, of course, satisfied by Lebesgue null
sets in Rn. Hence it seems natural in infinite-dimensional Banach spaces
to also look for a translation-invariant Borel measure p which would play
the role of the Lebesgue measure and then to consider its null sets. This
idea, however, does not pan out.
To see this, suppose for a contradiction that p is such a measure in a
separable infinite-dimensional Banach space X . As p is outer regular with
respect to open sets, there is an open U c X such that 0 < p ( U ) < 00.
Since X is infinite-dimensional, there are a number r > 0 and pairwise
disjoint balls B1,Bz, . . . of radius T inside U . As X is separable, there exist
balls El, &, . . . of radius r so that U c &. Hence 0 < p ( U ) 5 p ( g i ) ,
and since p is translation-invariant, we have
u
-
0 < p(B1) = p(B1) = p(B2) = p(B2) = .. ..
p(Bi) = m, which is indeed a contradiction.
Consequently, p ( U ) 2
This means that we need to define null sets differently, not using a single
measure. Here is a simple definition introduced by Christensen [4].
A Borel subset A of a separable Banach space X is called Huur nu12 if
there exists a Borel probability measure p on X so that p ( A x) = 0 for
all z E X .
A translate of a Haar null set is, clearly, Haar null. The probability
measure witnessing that a given set is null can be taken with its support
contained in an arbitrarily small ball. Hence open sets are not null. Suppose
that { A , } is a sequence of null sets, and that for each n the measure
p, witnessing that A, is null is a probability measure which in the weak
topology is “close enough” to the probability measure supported at the
origin. Then it is not difficult to see that the convolution p1 * p2 * p3 * . . .
witnesses that UA , is Haar null.
+
86
Haar null Borel sets coincide with Lebesgue null Borel sets in finitedimensional Banach spaces.
Christensen introduced this family to show that a Lipschitz function f
defined on a separable Banach space X is Giiteaux differentiable almost
everywhere. This means that it is differentiable outside a Haar null set,
and so, in particular, on a dense set.
Let us briefly sketch a proof of this, in order to understand how one uses
null sets and finite-dimensional results to prove density results in infinitedimensional spaces.
Let {u,} be a sequence of unit vectors dense in the unit sphere of X ,
and let A, be the normalized Lebesgue measure supported on the unit ball
of the span L, of (211,. . . , un}. Let A, c X be the set of all points z E X
such that f restricted to z + L, is not differentiable at z. By Rademacher’s
theorem, X,(A,
y) = 0 for all y E X . Hence each A, is Haar null. It is
not difficult to see that f is Giiteaux differentiable at every point outside
the Haar null set A = UA,.
There are many other notions of null sets in infinite-dimensional Banach
spaces. Aronszajn [ I ] and cube null sets [19] were introduced similarly
to Haar null sets in order to prove Giiteaux differentiablity of Lipschitz
functions, and later on turned out to be different definitions of the same
family of sets [5]. A generalization of Aronszajn null sets and the so-called
r-null sets [18] were introduced to examine the FrBchet differentiability of
Lipschitz functions. Rather than going into the sometimes quite technical
definitions of these famillies of null sets, we finish this section with a few
simple observations about Haar null sets, in order to give the reader some
more feeling for them.
Any closed proper (affine) subspace L and, in particular, any hyperplane, of a separable Banach space X is Haar null. To see this, it is enough
to take u E X \ L and the normalized Lebesgue measure X on [0,u].
Suppose K is a compact subset of an infinite-dimensional separable
Banach space X . As above, we will find a point u so that any line in the
direction of u intersects K in at most one point; the Lebesgue measure X
on [O,u]will show that K is Haar null. As K is compact, so is its closed
convex symmetric hull k.Since X is infinite-dimensional, the interior of
k is empty. Hence, as we have seen in the first section, by Baire’s theorem,
Unk # X.Now choose any u E X \ Ung.
It turns out [22] that in reflexive spaces all closed convex sets with
empty interior are Haar null; actually this is a characterization of separable
reflexive spaces. Even in Hilbert space, however, there is a convex set
+
87
which is Haar null, but this fact is not witnessed by any measure with a
finite dimensional support. To see this, let Q = { C a i e i E & : ai 2 0) be
the positive cone of &. Then Q is closed, convex and it has empty interior;
hence it is Haar null by [22]. At the same time, Q contains a translate of
the unit ball of every finite-dimensional subspace of l 2 .
Finally, let us mention that by far not all “good” properties of Lebesgue
null sets are shared by Haar null sets. If F : R” -+ R” is a Lipschitz
mapping, then the image of a Lebesgue null set is a null set again. There
exists, however, a mapping F of f$ onto itself such that both F and its
inverse are Lipschitz, and at the same time the complement of F ( A ) is
Haar null for some Haar null Bore1 set A c & [17].
4. Nonexpansive and contractive mappings
Recall that a Lipschiz mapping with Lipschitz constant equal to one
is said to be nonexpansive. Nonexpansive mapping theory has flourished
during the last forty years or so with many results and applications. See,
for example, [12], [13], [15], and the references mentioned therein. In contrast with the iterates of nonexpansive mappings which in general do not
converge, it is known that the iterates of contractive mappings (see the
definition below) converge in all complete metric spaces [23]. However, it
is also known [6] that in Banach spaces the iterates of most nonexpansive
mappings (in the sense of Baire’s categories) do converge to their unique
fixed points. In the next section we explain this result by showing that
most nonexpansive mappings are, in fact, contractive [26]. As a matter of
fact, it turns out that our result holds for all complete hyperbolic spaces,
a notion which is of independent interest and which we now recall.
Let (X, p ) be a metric space and let R denote the real line. We say that
a mapping c : R -+ X is a metric embedding of R into X if p ( c ( s ) ,c ( t ) )=
1s - t ( for all real s and t. The image of R under a metric embedding will be
called a metric line. The image of a real interval [a,b] = {t E R : a 5 t 5 b}
under such a mapping will be called a metric segment. Assume that (X,p )
contains a family M of metric lines such that for each pair of distinct points
x and y in X, there is a unique metric line in M which passes through x
and y. This metric line determines a unique metric segment joining x and
y. We denote this segment by [x,y]. For each 0 5 t 5 1, there is a unique
point z in [x,y] such that p(x,z ) = tp(x,y) and p(z, y) = (1 - t)p(z,y).
This point will be denoted by (1- t ) x @ ty. We will say that X, or more
aa
precisely ( X ,p, M ) , is a hyperbolic space if
1
2
1
2
p(-x @ -y,
1
1
1
63 )-.
2
2
I ,P(Y,
--2
.>
for all x , y and z in X . A set K c X is called pconvex if [z,y] c K for
all x and y in K . It is clear that all normed linear spaces are hyperbolic.
A discussion of more examples of hyperbolic spaces and, in particular, of
the Hilbert ball can be found, for example, in [24]. In the sequel we will
repeatedly use the following fact (cf. [13], pp. 77 and 104, and [24]): If
( X ,p, M ) is a hyperbolic space, then
p((1 - t ) x @ t z , (1 - t)?4@ t w ) I
(1 - t M x , Y)
+ M.,
w)
(1)
for all x , y , z and w in X and 0 5 t I
1.
Assume that ( X ,p) is a hyperbolic complete metric space and let K be a
bounded closed pconvex subset of X . Denote by A the set of all operators
A : K + K such that
p(Ax, Ay)
5 p(x, y) for all x,y
E
K.
(2)
In other words, the set A consists of all the nonexpansive self-mappings of
K . Set
d ( K ) = SUP{P(Z,Y)
:
x,y E K).
We equip the set A with the metric h(., .) defined by
h(A, B ) = sup{p(A~,Bz): z E K } , A, B E A.
Clearly, the metric space (A,h ) is complete.
We say that a mapping A E A is contractive if there exists a decreasing
function 4 A : [0,d ( K ) ]+ [0,1] such that
$A(t)< 1 for all t E (O,d(K)]
(3)
and
d A x , AY) 5 $A(P(GY))P(x, Y) for all 2 , y E K .
(4)
The notion of a contractive mapping, as well as its modifications and applications, were studied by many authors. See, for example, [16]. We now
quote a convergence result which is valid in all complete metric spaces [23].
Theorem 4.1. Assume that A E A is contractive. Then there exists X A E
K such that Anx -+ X A as n -+ 00, uniformly on K .
89
For each A, B E A and each a E (0, l ) , define the operator a A @ ( l - a ) B
by
( a A @ (1 - CX)B)Z
= AX @ (I - a ) B z , z E K.
(5)
Note that a A @ (1 - a ) B E A by (1). Next, we note the following simple
fact.
Proposition 4.1. If A E A is contractive, B E A and a E (0, l ) , then the
composition operators AB, B A and the p-convex combination a A @ (l - a ) B
are also contractive.
5. The set of contractive mappings contains a residual
subset
Now we show that most of the mappings in A (in the sense of Baire's
categories) are, in fact, contractive.
Theorem 5.1. There exists a set 3 which is a countable intersection of
open, dense sets in A such that each A E 3 is contractive.
Proof. Fix B E K . For each A E A and each y E (0, l ) , define A, E A by
A,z
=
(1 - ~ ) A @
z 70, 2
E
K.
(6)
Clearly, the set {A, : A E A, y E ( 0 , l ) ) is dense in A.
Let A E A and y E (0,l). Inequality (1) implies that
P(A-97 A,Y) I (1 - Y ) P ( G Y)
(7)
for all x,y E K . For each integer i 2 1, define
U(A, 7, i) = { B E A : h(A,, B ) < 4-2-lyd(K)}.
(8)
We will show that for each A E A, y E (0, l ) , and each integer i 2 1, the
following property holds:
( P l ) For each B E U ( A , y , i ) and each x , y E K satisfying p(x,y) 2
4-Zd(K), the inequality p ( B x ,By) 5 (1 - 2-'y)p(x, y) is valid.
Indeed, let A E A, y E (0,l ) , and let i 2 1 be an integer. Assume that
B
E
U ( A ,y,i),x,y E K , and p(x,y) 2 4-24K).
BY (8), (9), and (7),
p ( B x , BY) I
p(A,x, A,Y)
+ 2-'
. 4 - i y d ( K ) I (1 - Y)P(Z,Y)
(9)
90
Thus property (Pl) holds. Now define
3 = n g l U { U ( A ,y,q ) : A E A, y E (0,1)}.
Clearly, 3 is a countable intersection of open, dense sets in A. We claim
that any B E 3 is contractive. To prove this, assume that q is a natural
number. There exist A E A and y E ( 0 , l ) such that B E U(A, y,q). By
property (Pl), for each z,y E K satisfying p ( z , y ) 2 4 - 4 d ( K ) , we have
p(Bz,By) 5 (1 - 2-ly)p(z, g). Since q is an arbitrary natural number, we
can conclude that B is contractive. Theorem 5.1 is proved.
0
A variant of Theorem 5.1 has recently been established by De Blasi and
Pianigiani [lo] who used the Baire category method to solve the prescribed
singular values problem with Dirichlet boundary data. We now remark in
passing that an extension theorem of theirs, namely, Theorem 3.1 in [lo],
can be improved.
Theorem 5.2. Let H and L be two Halbert spaces and let D be a subset of
H . T h e n each contractive mapping A : D t L with a bounded range has a
contractive extension B : H -+ L to all of H .
Proof. Let A satsify (4) for all x,y E D,where the decreasing function
+ A : R+ 3 [0,1] satisfies (3) for all t > 0, and let v : R+ -+ R+ be its
modulus of continuity; that is, let
v ( t ):= S U ~ { ~ A-XAgl : Z, Y E D,)Z- Y)_< t } .
Clearly, v has a bounded range and v(t) 5 t for all t E R+.We claim that
v ( t ) < t for all positive t. Indeed, suppose v ( s ) = s for some s > 0. Then
there would be sequences {xn} and {yn} in D such that 12, - l,y 5 s for
all n E N and limnhco (Az, - Ay,I = s. Since
IA% - AYnI 5 $JA(I.n - Ynl)lzn - Ynl
I 1% - Ynl 5 s,
we conclude that (z, - yn( -+ s and that +A(Izn
- g,)) + 1 as n -+ m.
But this means that ( 2 , - yn) t 0, so that s = 0, which is a contradiction.
Thus v(t) < t for all t > 0, as claimed.
Now we can follow the argument in [lo] and obtain a concave function
w : R+ --f R+ such that w ( 0 ) = 0, v ( t ) 5 w ( t ) for all t 2 0, and w ( t ) < t for
t > 0. By the Grunbaum-Zarantonello extension theorem [14] (see also [2],
p. 18), A has a uniformly continuous extension B : H -+ L with modulus
of continuity o(t) 5 w ( t ) , t 2 0. Define $B : [0, m)
[0,1] by @(O) := 1,
$ B ( t ):= w ( t ) / t , t > 0. Then $B(t)< 1 for all t > 0 and $B is decreasing
--f
91
+
because w ( s ) 2 (1 - s/t)w(O) ( s / t ) w ( t ) = ( s / t ) w ( t ) , 0 5 s
concavityofw. Also, IBz-By1 I P(lz-yl) 5 w ( l ~ - y l ) = $ B ( l
for all z, y E H . Thus B is indeed contractive, as asserted.
< t , by the
~-~l)l~-~I
0
Note that if A : D 4 L satisfies / A x - Ayl < Iz - yI for z # y and D
is compact (as assumed in [ l o ] ) then
,
A is necessarily contractive. This,
however, is no longer true if D is not compact.
6. The complement of the set of contractive mappings is
a-porous
In the previous section we have explained the result in [6] by showing that
most nonexpansive mappings are, in fact, contractive. However, it is shown
in [7] that the complement of the set of power convergent nonexpansive
mappings is not only of the first category, but also a-porous. In this section we strengthen all of these results by showing that the set of all noncontractive mappings is not only of the first category, but also a-porous [28].
For simplicity, we will consider only self-mappings of a subset of a Banach
space. However, just as in the previous section, our result also holds for
the wider class of hyperbolic complete metric spaces.
Assume that (X, 1) . ) I ) is a Banach space and let K be a bounded closed
convex subset of X. We continue to use the notation of Section 4 with the
understanding that p(z,y) = 1111: - yI I.
Theorem 6.1. There exists a set 3 c A such that A \ 3 i s a-porous in
(A,h ) and each A E 3 is contractive.
Proof. For each natural number n, denote by A, the set of all A E A
which have the following property:
( P 2 ) There exists K E ( 0 , l ) such that IlAz - Ayll 5 1c11z- yII for all
z,y E K satisfying ) ) z- y J J_> d ( K ) ( 2 n ) - ' .
Let n 2 1 be an integer. We will show that the set A \ A, is porous in
(A,h). To this end, set
(u
= 8-' min{d(K), 1 } ( 2 n ) - l ( d ( K )
+ l)-l.
(10)
Fix 0 E K . Let A E A and r E ( 0 , 1 ] . Set
y = 2 - ' r ( d ( ~ )+ I)-'
(11)
and define
A,z
=
(1-y)Az
+ 70, z E K.
(12)
92
Clearly, A, E A,
w,,A) I 7 d ( K ) ,
and for all z, y E
(13)
K,
IIA+ - ArYll I
(1 - 7)llAz -AYll
I (1 - 7)llz - YII.
(14)
Assume that B E A and that
h(B,A,) 5 ar.
We will show that B E
Indeed, let
(15)
A.
z , y E K and I(z-
YII
2 (2n)-'d(K).
(16)
It follows from (14) and (16) that
1111: - YII
- IIA,z
-
(17)
ArYll 2 Y l l Z - YII L 7 4 K ) ( W - l .
BY (15),
IIBz - BY11 I
IlBz - A+Il
+ I I 4 z - A,Yll+ I P , Y
I llA+ - A,yll+
2ar.
When combined with (17), (11) and
113: - YII -
llBz - B Y l l 2
- BY11
113: - YII
(lo), this implies that
- IlA+ - A,Yll - 2 a r
2 yd(K)(2n)-'
- 2ar
= 2-'r[(2n)-'d(K)(d(K)
2 2T1rd(K)(4n)-'(d(K)
+ 1)-' - 4a]
+ 1)-l.
Thus
JJBx- By11 I
1
1
2-yJJ
-
+
r d ( K ) ( d ( K ) 1)-'(8n)-'
I llz - Yll(1- .(8n)-'(d(K)
+ I)-').
Since this holds for all z,y E K satisfying (16), we conclude that B E
A,. Thus each B E A satisfying (15) belongs to A,. In other words,
{ B E A : h(B,A,) 5 o r } C A,.
(18)
If B E A satisfies (15), then by (13), ( l o ) , and (11) we have
h(A, B ) 5 h ( B ,A,)
+ h(A,,
A ) 5 ar + y d ( K ) 5 8-'r
+ 2T'r
Thus
{ B E A : h ( B ,A,)
I a r } C { B E A : h(B,A) I r } .
5 r.
93
When combined with (18), this inclusion implies that A \ A, is porous in
(A,h). Set .F = n,",ld,.
Clearly, A \ .F is a-porous in (A,h). In view of
property (P2), each A E F is contractive.
0
Remark. If X is a Hilbert space, then the set of all strict contractions
(that is, mappings with Lipschitz constant strictly less than one) is a-porous
in (A,h ) [7]. It would be of interest to determine if this continues to hold
in all Banach spaces, as well as for rotative mappings [15], Chapter 11.
Analogous results to Theorem 6.1 and to the above remark for setcontractions with respect to Kuratowski's measure of noncompactness were
established in [3].
7. Well-posedness
In this section we show that the fixed point problem for any contractive
mapping is well posed [31]. Therefore Theorem 6.1 also yields Theorem 8
in [7].
Let ( K ,p) be a bounded complete metric space. We say that the fixed
point problem for a mapping A : K
K is well posed if there exists a
unique X A E K such that A X A= X A and the following property holds:
if {x,},",~ c K and p(xn,Ax,) -+ 0 as n + 03, then p(x,, X A )
0 as
---f
---f
n
4
03.
The notion of well-posedness is of central importance in many areas of
mathematics and its applications. In our context this notion was studied in
[7], where generic well-posedness of the fixed point problem is established
for the space of nonexpansive self-mappings of K .
We continue to use the notation introduced in Section 4.
Theorem 7.1. Assume that a mapping A : K
the fixed point problem for A is well posed.
-+
K is contractive. Then
Proof. Since the mapping A is contractive, there exists a decreasing function qbA = qb : [O,d(K)]-+ [0,1] such that (3) and (4) hold. By Theorem 4.1,
there exists a unique X A E K such that
A X A= X A .
Let {x,},"=~
c K satisfy
lim p(x,,Axn) = 0.
n-oo
(19)
94
We will show that x, -+ X A as n + 00. Suppose this were false. By
extracting a subsequence, if necessary, we might then assume without loss
of generality that there exists E > 0 such that
p(x,,xA) 1 E for all integers n 2 1.
(21)
It follows then from (19), (4), (21) and the monotonicity of the function 4
that for all integers n 2 1,
p(xA,Zn) _< P(ZA,Azn) +p(A%,%)
I p(Axn, zn) + 4(p(z,,
I p(Azn, 2 , )
ZA))P(G,
%A)
(22)
+ +(E)P(ZA,xn).
Inequalities (22) and (21) imply that for all integers n 2 1,
E(1
- 4(E))
5 (1- 4(E))P(xA,xn) 5 p(Axn,xn),
a contradiction (see (20)). The contradiction we have reached proves Theorem 7.1.
n
8. Strict contractions
Let K be a nonempty, bounded, closed and convex subset of a Banach space
( X ,II . 11). Set
rad(K) = sup(llx1l : x E K } .
For each A : K
-+ X
(23)
, let
LiP(A) = sup{llAz
-
AYIlAb - YII :
Z,Y E
K , x # Y}
(24)
be the Lipschitz constant of A. We continue to denote by d the set of all
nonexpansive mappings A : K -+ K , that is, all self-mappings of K with
Lip(A) 5 1, or equivalently, all self-mappings of K which satisfy
llAx - Ayll 5 llx - yII for all x , y E K .
(25)
We say that a self-mapping A : K -+ K is a strict contraction if Lip(A) < 1.
We have already seen that although the set of all strict contractions in
is small, the set of power convergent nonexpansive mapthe space (d,h)
pings is large.
In Sections 5, 6 and 7 we have singled out a property (that of being
contractive), which on the one hand turns out to be possessed by most
nonexpansive mappings and on the other ensures power convergence to
unique fixed points as well as well-posedness of the fixed point problem.
95
It may seem strange that the set of contractive mappings is large while
the set of strict contractions is small. Needless to say, the answer to the
question whether a certain set is small or large depends on the metric
(topology) under consideration. Indeed, in this section we show that if the
space A is endowed with another natural complete metric, then the set of
strict contractions is, in fact, large [32]. Our new metric is defined by
d(A,B) = sup{llAz - BzlJ : z E K } +Lip(A - B ) ,
(26)
where A, B E A. It is not difficult to see that the metric space (A,d) is
complete.
Theorem 8.1. Denote by 3 the set of all strict contractions A E A. Then
A \ 3 is porous.
Proof. Fix a number
Q
> 0 such that
Q < (1 + 2rad(K))-l32-l
(27)
and fix 0 E K . Let A E A and let r E (0,1]. Set
y = (1+ 2rad(K))-lr/S
(28)
and put
A,z=(l-y)Az+yB,
XEK.
(29)
I
(1 - 7)Ilz - YII.
(30)
Clearly, A, E A and for each z, y E K ,
IlA+
- 4 V l l = (1 - r)llAz - AYll
By (29), (23), (24) and (26), for each z E K ,
I J A ,~ A ~ J=I11(i- r ) ~ z+ y e
I2yrad(K),
- ~ z l=
l rile -~ z l l
96
We now assume that B E A and t h a t
d(B,A,) 5 ar.
(33)
In view of (33), (26), (30) and (28), we see that
Lip(B) ILip(A,)
ILip(A,)
+ Lip(B - AY) 5 Lip(A,) + d ( B , A,)
+ ar 5 (1 - y) + ar I 1 - (r/8)(1+2rad(K))-'
+ r(32(1+ 2rad(K))-l
and so B E
I
1 - (r/16)(1+ 2rad(K))-l < 1,
F. Clearly, by (33), (32) and
(27),
+
d(B, A ) 5 d ( B , AY) d(A,, A) I ar
Thus for each B E A satisfying (33), B E
completes the proof of Theorem 8.1.
F
+ r/8 I r.
and d ( B , A ) 5 r. This
0
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99
ABSOLUTE-VALUED ALGEBRAS, AND
ABSOLUTE-VALUABLE BANACH SPACES *
ANGEL RODR~GUEZPALACIOS
Universidad de Granada, Facultad de Caencias,
Departamento de Andlisis Matema'tico, 180'71-Granada (Spain)
E-mail: [email protected]
Absolute-valued algebras are fully surveyed. Some attention is also payed to Bai
nach spaces underlying complete absolute-valued algebras.
Introduction
Absolute-valued algebras are defined as those real or complex algebras A
satisfying 11zy/11 = IIzlIIIy/I] for a given norm 1) . 11 on A , and all z,y E A .
Despite the nice simplicity of the above definition, absolute-valued algebras have not attracted the attention of too many people. A reason could
be that, in the presence of associativity, the axiom (1xyll = llxllIlyll is extremely obstructive. Indeed, according to an old theorem of S. Mazur 66,
there are only three absolute-valued associative real algebras. Nevertheless,
when associativity is removed, absolute-valued algebras do exist in abundance. Some facts corroborating the above assertion are that every complete
normed algebra is isometrically algebra-isomorphic to a quotient of a complete absolute-valued algebra (Corollary 3.2), and that every Banach space is
linearly isometric to a subspace of a complete absolute-valued algebra (Theorem 5.1). Anyway, the quantity and quality of works on absolute-valued
algebras seemed to us enough to deserve a detailed survey paper like the
one we are just beginning.
Our paper collects the results on absolute-valued algebras since the pioneering works of Ostrowski 7 5 , Mazur 66, Albert 3 , and Wright log (see
Subsection 1.3) to the more recent developments. The inflexion point in the
'
'This work is partially supported by Junta de Andalucia grant FQM 0199 and Projects
I+D MCYT BFM2001-2335 and BFM2002-01810
100
theory, namely the Urbanik-Wright paper lo6,is fully reviewed (see Subsections 2.1, 2.2, 2.3, and 3.1). Among the recent developments, we emphasize
the solution 57 t o Albert's old problem if every absolute-valued algebraic
algebra is finite-dimensional (see Subsection 2.7), and the study of Banach
spaces underlying complete absolute-valued algebras, done in and 69 (see
Section 5). A special attention is also payed to the intermediate works of
K. Urbanik (lol to lo4) and M. L. El-Mallah (35 to 46). This is done in
Subsections 2.4, 2.5, 2.7, 3.1, 3.2, and 3.4. Contributions of other authors
(including the one of this paper) are also reviewed (see mainly Subsections 1.4, 2.6, 3.5, and 3.6, and the whole Section 4). The clarifications of
the theory at some precise points, done by Gleichgewicht 49 and ElduqueP6rez 331 are inserted in the appropriate places (see Subsection 3.4, and
Subsections 1.3, 2.1, and 3.5, respectively). Our paper contains also some
new results, and several new proofs of known results. Known proofs have
been included only when they seemed to us specially illuminating.
As far as we know, absolute-valued algebras have been surveyed in exclusive several times (see 86, 'l, and 'lo), but in references not easily available,
and never in English. Moreover, references 91 and 'lo are relatively short,
and references and 'lo become today rather obsolete. On the other hand,
there are also survey papers on more general topics, devoting to absolutevalued algebras some attention (see 87 and 8 8 ) . Finally, let us note that
the Ph. Theses 35, 63, and 78 are devoted to absolute-valued algebras, and
contain both reviews of other people's results and proofs of results of their
respective authors.
1. Finite-dimensional absolute-valued algebras
1.1. Some basic definitions and facts
By an algebra over a field F we mean a vector space A over F endowed
with a bilinear mapping (x, y) -+ xy from A x A to A called the product
of the algebra A . Algebras in this paper are assumed t o be nonzero, but
are not assumed to be associative, nor to have a unit element. We suppose
that the reader is familiarized with the basic terminology in the theory
of algebras. Thus, terms as subalgebra, ideal, or algebra homomorphism
are not defined here. For an element x in an algebra A, we denote by L ,
(respectively, R,) the operator of left (respectively, right) multiplication
by x on A. The algebra A is said to be a division algebra if, for every
nonzero element 2 of A, the operators L, and R, are bijective. An algebra
is said t o be alternative if it satisfies the identities xfx2 = x1(x1x2) and
101
( 2 1 2 2 ) 2 2 = x ~ x g . We note that alternative algebras are “very nearly”
associative. Indeed, by Artin’s theorem (see Theorem 2.3.2 of ‘13), the
subalgebra generated by two arbitrary elements of an alternative algebra
is associative. It is also worth mentioning that every alternative division
algebra has a unit (see page 226 of 31). By an algebra involution on an
algebra A we mean an involutive linear operator x -+ x* on A satisfying
(zy)* = g*x* for all x , y E A .
Now, let K denote the field of real or complex numbers. An algebra norm (respectively, absolute value) on an algebra A over K is a
norm 11 . I( on the vector space of A satisfying llxyll 5 IIxlI ((yI((respectively,
llzy[l = Ilxl[[[y[()
for all x , y E A . By a normed (respectively, absolutevalued) algebra we mean an algebra over IK endowed with an algebra
norm (respectively, absolute value). We note that absolute-valued j n i t e dimensional algebras are division algebras. We also note that, if there exists an absolute value o n a finite-dimensional algebra A over K,then we
can speak about “the” absolute value of A , understanding that such an
absolute value is the unique possible one o n A . This is a straightforward
consequence of the easy and well-known result immediately below. The
proof we are giving here is taken from 26.
Proposition 1.1. Let A be a normed algebra over K,let B be an absolutevalued algebra over K,and let $ : A + B be a continuous algebra homomorphism. T h e n $ is in fact contractive.
Proof. Assume to the contrary that q5 is not contractive. Then we can
choose a norm-one element z in A such that Il$(x)l[> 1. Defining inductively 2 1 := x and x,+1 := xi, we have 11$(x,)1) = 11q5(x)112n-14 00 .
Since Ilxnll 5 1, this contradicts the assumed continuity of $.
0
Looking at the above proof, we realize that Proposition 1.1remains true
if B is only assumed to be a normed algebra over K satisfying ((y2((
=( ( ~ 1 1
for every y E B , and 4 : A -+ B is only assumed to be a continuous linear
mapping preserving squares. As a consequence of Proposition 1,1, every
continuous algebra involution o n a n absolute-valued algebra is isometric.
Let A be a normed algebra. An element z of A is said to be a left
(respectively, right) topological divisor of zero in A if there exists a
sequence { x n } of norm-one elements of A such that {xz,} 4 0 (respectively, {z,x} + 0). Elements of A which are left or right (respectively,
both left and right) topological divisors of zero in A are called one-sided
~
102
(respectively, two-sided) topological divisors of zero in A. The element IC E A is said to be a joint topological divisor of zero in A if there
exists a sequence {zn}of norm-one elements of A such that { x z n } -+ 0 and
{xnx} -+ 0. We note that both absolute-valued algebras and normed division
alternative algebras have no one-sided topological divisor of zero other than
zero. (This is clear in the case of absolute-valued algebras, and is easily
verified in the case of normed division alternative algebras, by keeping in
mind the fact already pointed out that division alternative algebras have a
unit element, and applying the properties of “invertible” elements of unital
alternative algebras given in page 38 of 94.) We will review in Theorem 1.1
a much deeper fact implying that, conversely, normed alternative algebras
without nonzero joint topological divisors of zero are division algebras.
1.2. Quaternions and Octonions
Surveying absolute-valued algebras, we should write something about the
algebra W of Hamilton’s Quaternions, and the algebra 0 of Octonions
(also called “Cayley numbers”). These algebras, together with the fields
of real and complex numbers (denoted by R and C, respectively), become
the basic examples of absolute-valued algebras. The algebras C, W,and 0
can be built from R by iterating the so-called “Cayley-Dickson doubling
process” (see for example pages 256-257 of 31). Thus, if A stands for either
R, @, or MI, and if * denotes the standard algebra involution of A (which, for
A = R, is nothing other than the identity mapping), then we can consider
the real vector space A x A with the product given by
(21,52)(2.3,24)
:= (51x3 - - 4 x a , x T 2 4
+
23x2)
7
obtaining in this way a new real algebra which is a copy of either 6,H,
or 0,
respectively. In this doubling process, the standard involution * and
the absolute value 11 . I( of the new algebra are related to the corresponding
ones of the starting algebra by the formulae
respectively. It follows from the last formula that the absolute values of
R, C, W , and 0 come from inner products. It is also of straightforward
verification that the algebra W is associative but not commutative, whereas
the algebra 0 is alternative but not associative.
The joint introduction of IHI and 0done above is surely the quickest possible one. However, concerning W, there is another more natural approach.
103
Indeed, in the same way as C can be rediscovered as the subalgebra of the
algebra M2(R) (of all 2 x 2 matrices over R) given by
W can be rediscovered as the real subalgebra of Mz(C)given by
(see for example page 195 of 3 1 ) . Regarded W in this new way, the standard
involution of W corresponds with the transposition of matrices, and the
absolute value of an element of W is nothing other than the nonnegative
square root of its (automatically nonnegative) determinant.
The algebras MI and 0 are very far from being only exotic objects in
Mathematics. By the contrary, they solve many natural problems in the
field of the Algebra, the Geometry, and the Mathematical Analysis. Thus,
as a consequence of the Robenius-Zorn theorem (see for example pages 229
and 262 of 3 1 ) , R, C, MI, and 0 are the unique finite-dimensional alternative
division real algebras. On the other hand, we have the following.
Theorem 1.1. Every normed alternative real algebra without nonzero joint
topological divisors of zero is algebra-isomorphic to either R, C, W,or 0.
Theorem 1.1has been proved by M. L. El-Mallah and A. Micali 45 by applying the forerunner of I. Kaplansky 6o (see also 17) for associative algebras.
Keeping in mind the uniqueness of the absolute value on a finite-dimensional
algebra, pointed out in Subsection 1.1, it follows that R, C, W,and 0 are
the unique absolute-valued alternative real algebras. A refinement of the fact
just formulated (see Theorem 2.4) and other interesting characterizations
of R, C, W,and 0 (see Theorems 2.1, 2.5, 2.6, and 3.4) will be reviewed
later. The reader interested in increasing his knowledge on Quaternions
and Octonions is referred to the books 22 and 31, and the survey papers
and 98. These works and references therein will provide him with a complete
panoramic view of the topic. Nevertheless, let us emphasize the abundance
of historical notes and mathematical remarks collected in 31, and take some
samples of them.
Thus, in a note written with the occasion of the fifteenth birthday of
the Quaternions, W. R. Hamilton says: “They [the Quaternions] started
into life, or light, full grown, on the 16th of October, 1843, as I was walking
with Lady Hamilton to Dublin, and came up to Brougham Bridge.” (see
104
page 191 of 3 1 ) . It turns out also curious to know that Hamilton tried for
many years to built a three dimensional division associative real algebra. In
fact, shortly before his death in 1865 he wrote to his son: “Every morning,
on my coming down to breakfast, you used to ask me: ‘Well, Papa, can
you multiply triplets?’ Whereto I was always obliged to reply, with a sad
shake of the head: ‘No, I can only add and subtract them’.’’ (see page 189
of 31). It is not less curious how, in a very elemental way, one can realize
that the attempt of Hamilton just quoted could not be successful. Indeed,
refining slightly the content of the footnote in page 190 of 3 1 , we have the
following.
Proposition 1.2. Let A be a (possibly nonassociative) division real algebra
of odd dimension. Then A has dimension 1, and hence it is isomorphic to R.
A\ {0}, and let x be in A. Then the characteristic polynomial of the operator LY1 o L, must have a real root (say A), which becomes
Proof. Fix y
E
an eigenvalue of such an operator. Taking a corresponding eigenvector
z # 0, we have (z - Xy)z = 0, which implies z = Xy. Since 2 is arbitrary
in A, we have A = R y .
0
We note that the above proof is nothing other than a natural variant of
the usual one for the fact that finite-dimensional division algebras over an
algebraically closed field IF are isomorphic to IF.
According to the information collected in page 249 of 31, the Octonions
were discovered by J. T. Graves in December 1843, only two months after the birth of the Quaternions. Graves communicated his discovery to
Hamilton in a letter dated 4th January 1844, but did not publish it until
1848. In the meantime, just in 1845, the Octonions were rediscovered by A.
Cayley, who published his result immediately. For a more detailed history
of the discovery of Octonions the reader is referred to pages 146-147 of 6.
1.3. The pioneering work of Ostrowski, Mazur, Albert, and
Wright
We already know that R, @, W,and 0 are the unique absolute-valued
alternative real algebras. As a consequence, B,@, and W are the unique
absolute-valued associative real algebras (a fact first proved by S. Mazur 66).
More particularly, we have the following.
Proposition 1.3. Let A be a n absolute-valued, associative, and commutative algebra over R. Then A is isometrically isomorphic to either R or @.
105
Proof. Since A is an integral domain, we can consider the field of fractions
of A (say IF), and extend (in the unique possible way) the absolute value of
A to an absolute value on IF. Now IF is an absolute-valued field extension
of R,and hence it is isometrically isomorphic to R or C (see Lemma 1.1
below). Since A is a subalgebra of IF, the result follows.
0
According t o the information collected in pages 243 and 245 of 31, the
above proposition and proof are due t o A. Ostrowski 75, who seems to have
been the first mathematician considering absolute-valued algebras as abstract objects which are worth being studied. The following lemma (today
a consequence of the famous Gelfand-Mazur theorem) is also due t o him.
Lemma 1.1. Every absolute-valued field extension of
isomorphic to either R or @.
R is isometrically
The first paper dealing with absolute-valued algebras in a general nonassociative setting is the one of A. A. Albert 2 , who proves as main result the
following.
Proposition 1.4.R,C, W,and 0 are the unique absolute-valued finitedimensional real algebras with a unit.
A surprisingly short proof of Proposition 1.4, based on early works of
H. Auerbach and A. Hurwitz 5 3 , can be given. However, since such a
proof was not noticed by Albert, nor by anybody at his time, we prefer
to postpone it until the conclusion of Subsection 2.2, and continue here
with the chronological narration of facts. As we will see in Proposition 1.6
below, Proposition 1.4 was refined shorty later by Albert himself. Thus,
the actual interest of Albert’s paper relies on both the introduction of the
notion of “isotopy” between absolute-valued algebras, and the proof of the
following proposition.
Proposition 1.5. Let A be a n absolute-valued finite-dimensional real algebra. Then A is isotopic to either R,C, W,or 0. Therefore A has dimension
1 , 2 , 4 , or 8, and the absolute-value of A comes from a n inner product.
According to Albert’s definition, two absolute-valued algebras A and
B over K are said t o be isotopic if there exist linear isometries 4 1 , 4 2 , 4 3
from A onto B satisfying & ( z y ) = # Q ( x ) ~ ~ for
( Y )all z , y in A . Albert
derives Proposition 1.5 from Proposition 1.4 in a clever but quite simple
way. Indeed, choosing a norm-one element a E A , and defining a new
product 0 on the normed space of A by s a y := R;’(z)L;’(y), we obtain
106
a finite-dimensional absolute-valued algebra, which is isotopic to A and
has a unit (namely, a 2 ) . The argument just reviewed has been recently
refined in the paper of A. Elduque and J. M. PQrez 33, yielding Lemma 1.2
immediately below. As we will see later, such a lemma has turned out to
be very useful in the theory.
Lemma 1.2. Let A be a n absolute-valued algebra over K such that there
exist a , b E A satisfying a A = A b = A . Then A is isotopic to an absolutevalued algebra over K having a unit element.
Proof. We may assume that llall = llbll = 1. Then, defining a new product
0 on the normed space of A by z 0 y := R b 1 ( z ) L ; ' ( y ) , we obtain an
absolute-valued algebra over K, which is isotopic to A, and has a unit
0
(namely, ab).
Concerning the assertion in Proposition 1.5 about the dimension of
absolute-valued finite-dimensional real algebras, it is worth mentioning
that, some years after Albert's paper (just in 1958), it was proved the
following.
Theorem 1.2. Every finite-dimensional division real algebra has dimension 1 , 2 , 4 , or 8.
The paternity of Theorem 1.2 seems to be rather questioned. Indeed,
according t o 31, 48, and 6 , such a theorem was first proved by Kervaire
and Milnor 6 8 , Adams l, and Kervaire 62 and Bott-Milnor 13, respectively.
Anyway, in contrast with the case of Proposition 1.2, all known proofs of
Theorem 1.2 are extremely deep.
A second paper of Albert contains as main result the following refinement of Proposition 1.4.
Proposition 1.6. Let A be a n absolute-valued algebraic real algebra with
a unit. Then A is equal to either R, C , W,or 0.
We recall that an algebra A is called algebraic if all single-generated
subalgebras of A are finite-dimensional. As we will see later, Proposition 1.6
has been also refined, in two different directions, and at two very distant
dates (see Theorems 2.1 and 2.11). Therefore, Proposition 1.6 has today
the unique interest of having been, some years later, one of the key tools in
the original proofs of more relevant results in the theory of absolute-valued
algebras. Among these results, we limit ourselves for the moment to review
the one of F. B. Wright log which follows.
107
Theorem 1.3. An absolute-valued algebra over i
K i s finite-dimensional i f
(and only i f ) it i s a division algebra.
Albert’s paper also contains the particular case of Theorem 1.3 that
absolute-valued algebraic division algebras are finite-dimensional. However,
the proof given in for this result seems to us not to be correct. To
conclude the present section, let us note that Propositions 1.3 and 1.6, and
Theorem 1.3 above become “ap6ritifs” for Section 2 below.
1.4. Classification
A,
For A equal to either C, W, or 0,
let us denote by
*A,and A* the
absolute-valued real algebras obtained by endowing the normed space of
A with the products x @ y := x*y*, x 0 y := x * y , and x @ y := x y * ,
respectively, where * means the standard involution. It follows easily from
*
Proposition 1.5 that C, C,*C,and C* are the unique absolute-valued twodimensional real algebras. Therefore, to be provided with a classification
(up to algebra isomorphisms) of all finite-dimensional absolute-valued real
algebras, it would be enough to obtain such a classification in dimension
4 and 8. Whereas for dimension 8 the problem seems to remain open, the
case of dimension 4 has been solved in the paper of M. I. Ramirez 77, by
applying Proposition 1.5 and the description of all linear isometries on W
(see page 215 of 31). To this end, the so-called principal isotopes of
W are considered. These are the absolute-valued real algebras W l ( a ,b),
W z ( a ,b), W3(a, b), and &(a, b) obtained from fixed norm-one elements a , b
in W by endowing the normed space of W with the products x 0y := axyb,
z 0y := ax*y*b, x @ y := x*ayb, and x 0y := axby*, respectively. Then it
is proved the following.
Proposition 1.7. E v e y four-dimensional absolute-valued real algebra is
isomorphic to a principal isotope of W. Moreover two principal isotopes
W i ( a ,b) and Wj(a’, b’) of H are isomorphic if and only if i = j and the
equalities a‘p = &pa and b‘p = Spb hold for some norm-one element p E W
and some E , S E (1, -1).
Proposition 1.7 can be also derived from ”. A refinement of it can
be found in 19. The paper 77 also contains a precise description of all
four-dimensional absolute-valued real algebras with a left unit, as well as
many examples of four-dimensional absolute-valued algebras containing no
two-dimensional subalgebra.
108
Eight-dimensional absolute-valued real algebras with a left unit have
been systematically studied in the recent paper of A. Rochdi 79. As a first
basic result, Rochdi proves the following.
Proposition 1.8. The finite-dimensional absolute-valued real algebras with
a left unit are precisely those of the form A,, where A stands for either R,
@, IHI, or 0, p : A -+ A is a linear isometry fixing 1, and A, denotes the
absolute-valued real algebra obtained by endowing the normed space of A
with the product x 0 y := cp(x)y. Moreover, given linear isometries cp, 4 :
A 4 A fixing 1, the algebras A, and A4 are isomorphic i f and only if there
exists an algebra automorphism $ of A satisfying 4 = $ 0 cp 0 $-’.
It is proved also in 79 that, for A and cp as in Proposition 1.8, subalgebras of A, and cp-invariant subalgebras of A coincide. Moreover, a
linear isometry p : 0 -+ 0 fixing 1 can be built in such a way that 0
has no four-dimensional pinvariant subalgebra. It follows that there exist
eight-dimensional absolute-valued real algebras with a left unit, containing
no four-dimensional subalgebra. Such algebras are characterized, among
all eight-dimensional absolute-valued real algebras with a left unit, by the
triviality of their groups of automorphisms. Such algebras seem to become
the first examples of eight-dimensional division real algebras containing no
four-dimensional subalgebra.
In Subsection 3.4 we will review in detail the results concerning those
absolute-valued real algebras A endowed with an isometric algebra involution * which is different from the identity operator and satisfies xx* = x*x
for every x E A. In the finite-dimensional case, such algebras have been
classified in
The classification theorem has a flavour similar to that of
Proposition 1.8.
Right Moufang algebras are defined as those algebras satisfying the
identity x ~ ( ( x I x ~ =
) x(~( )x ~ x I ) x ~ Absolute-valued
)x~.
right Moufang algebras are considered by J. A. Cuenca, M. I. Ramirez, and E. SBnchez 2 4 ,
who show that such algebras are finite-dimensional. More precisely, they
prove Theorem 1.4 immediately below. The formulation of such a theorem
involves the notation introduced in Proposition 1.8 above, as well as the
result of N. Jacobson 55 that both IHI and 0 have an “essentially” unique
involutive automorphism different from the identity operator.
Theorem 1.4. The absolute-valued right Moufang real algebras are R, @,
IHI, 0,*@, and the algebras A,, where A stands for either JHI or 0, and cp
denotes the essentially unique involutive automorphism of A different from
109
the identity operator.
2. Conditions on absolute-valued algebras leading to the
finite dimension
2.1. The noncommutative Urbanik- Wright theorem
Despite the constant scarcity of works on absolute-valued algebras along
the history, a relatively short paper of K. Urbanik and F. B. Wright lo6,
appeared in 1960 and announced the same year in lo5, attracted the attention of many people because of the nice simplicity of its powerful results.
In fact, Urbanik-Wright theorems have become the key tools in the later
development of the theory of absolute-valued algebras. The first surprising
result in the Urbanik-Wright paper is the following.
Theorem 2.1. For an absolute-valued real algebra A , the following conditions are equivalent:
(1) There exists a E A \ (0) satisfying a x = xu, a ( a x ) = a2x, and
(xu). = xu2 for evenJ x E A.
(2) A has a unit element.
(3) A is equal to either R,@, W,or 0.
We shall call the crucial implication (2) 3 (3) in Theorem 2.1 above
the noncommutative Urbanik-Wright theorem. Such a theorem
immediately “works havoc” in the theory. For instance, it follows from
it, and Albert’s ideas about isotopes, that an absolute-valued algebra A
over R is finite-dimensional if (and only i f ) there exists a E A satisfying
a A = Aa = A. This refinement of Wright’s Theorem 1.3 attains a better form whenever Lemma 1.2 replaces Albert’s ideas. Thus we have the
following.
Theorem 2.2. An absolute-valued algebra A over K is jinite-dimensional
if (and only i f ) there exist a, b E A satisfying a A = Ab = A .
Even, applying an easy argument of completion (see 26 for details), we
derive from Theorem 2.2 a still better form of Theorem 1.3. Indeed, an
absolute-valued algebra A is finite-dimensional i f (and only i f ) there exist
a, b E A such that a A and Ab are dense in A. Theorem 2.2 was first proved
by the author 85 with other techniques. The proof given here is taken from
the Elduque-P6rez paper 33.
110
2.2, Kaplansky 's prophetic proof of the noncommutative
Urbanik- Wright theorem
Concerning the proof of the noncommutative Urbanik-Wright theorem, the
interested reader could go into the original paper lo6to see how Urbanik and
Wright apply, to commutative subspaces, Schoenberg 's characterization 95
of pre-Hilbert spaces as those normed spaces X satisfying
1
1
5
+ Y1I2 +
1
1
2-
Yll 2 2 4
for all norm-one elements z , y E X (see Remark 2.1 later), and how then,
after some technical arguments, they show that the algebra satisfies the
requirements in Albert's Proposition 1.6. However, it seems to us more
instructive to sketch how a proof of the noncommutative Urbanik-Wright
theorem can be tackled by the light of the present knowledge.
Actually, the proof of the noncommutative Urbanik-Wright theorem can
be divided into two parts. The first one, of a purely analytic type, consists in
realizing that absolute-values on unital algebras come from inner products.
This question was completely clarified twenty years ago. Indeed, it is easy
to show that unital absolute-valued algebras become particular cases of the
so-called smooth-normed algebras (see the proof of ( b ) + ( a ) in Corollary 29
of 82), and it follows from Theorem 27 of 82 that the norm of every smoothnormed algebra derives from an inner product (see also Section 2 of 84 for a
considerable simplification of the arguments in 82). We recall that a normed
space X over K is said to be smooth at a norm-one element z E X if the
closed unit ball of X has a unique tangent real hyperplane at z, and that
smooth-normed algebras are defined as those normed algebras A over
K having a norm-one unit 1 such that the normed space of A is smooth
at 1. Incidentally, we note that C is the unique smooth-normed complex
algebra, and that R, C, W,and 0 are the unique smooth-normed alternative
real algebras (see 82 and 84, and references therein). We also remark that
other arguments of more autonomous nature, showing as well that unital
absolute-valued algebras are pre-Hilbert spaces, have been found later by
El-Mallah 40 and the author 85 (see Theorems 3.2 and 3.5, respectively).
Now that we know that absolute values on unital algebras over K derive
from inner products, the second (and last) part of the suggested proof of the
noncommutative Urbanik-Wright theorem (now of a purely algebraic type)
begins with an easy observation. Indeed, if an absolute value o n a (possibly
nonunital) real algebra A comes from an inner product, then we are provided with a nondegenerate quadratic form q o n A (namely, the mapping
z -+ llzl12)satisfying q ( s y ) = q ( z ) q ( y )for all z, y E A. In this way, we nat-
111
urally meet the so-called composition algebras, and the problem of classifying them. This problem was already considered and solved by Hurwitz 53
under the additional requirements of finite dimension and existence of a
unit. Later Kaplansky “ proved that the assumption of finite dimension
in Hurwitz’s theorem is superfluous (see also Chapter 2 of ‘13). Applying
the Hurwitz-Kaplansky theorem, we obtain that the unique unital composition real algebras are R, C, R2 (with coordinate-wise multiplication),
MI, M2(R), 0,
and a certain eight-dimensional alternative nonassociative
algebra 0’
which (as for the case of R2 and M2(R)) has nonzero divisors of
zero. Since this last pathology is prevented in the case of absolute-valued
algebras, the proof of the noncommutative Urbanik-Wright theorem is then
concluded.
In the paper
just quoted, which was published seven years before
the one of Urbanik and Wright, Kaplansky prophesies both the noncommutative Urbanik-Wright theorem and a proof similar to that we have
sketched above. Even, it seems that he thinks that the noncommutative
Urbanik-Wright theorem was already proved at that time. Thus, he says
that “Wright log succeeded in removing the assumption [in Albert’s Proposition 1.61 that the algebra is algebraic”. Since we know that the above
assertion is not right, we continue reproducing Kaplansky’s words with
the appropriate corrections and explanations: “Wright proceeds by proving that the norm [of a unital absolute-valued DIVISION algebra] springs
from an inner product [see Lemma 3.2 later], and then that the algebra is
algebraic. ... Thus Albert’s finite-dimensional theorem [i.e., Proposition
1.41 can be proved by combining Wright’s result with Hurwitz’s classical
theorem on quadratic forms admitting composition [see also Proof of Proposition 1.4 below]”. Immediately, Kaplansky motivates his work by saying
that “The main purpose of this paper is to make a similar method possible
in the infinite-dimensional case by providing a suitable generalization of
Hurwitz’s theorem.”
Concerning the proof of the noncommutative Urbanik-Wright theorem
just sketched, let us also comment that, really, the two parts in which we
have divided it overlap somewhat. This is so because the proofs of the
results in 82, 40, and 85, implying that unital absolute-valued algebras are
pre-Hilbert spaces, give simultaneously a rich algebraic information, which
is also provided by a part of the proof of the Hurwitz-Kaplansky theorem. In fact, with such an additional information in mind, the proof of the
noncommutative Urbanik-Wright theorem can be concluded by applying
the Frobenius-Zorn theorem instead of the one of Hurwitz-Kaplansky (see
112
Remark 31 of s2 and Remark 4 of s5 for details).
To conclude the present subsection, let us show how actually Albert
could have derived his Proposition 1.4 from Hurwitz’s theorem, if he were
aware of a result of Auerbach (see also Theorem 9.5.1 of ”) implying that
finite-dimensional transitive normed spaces are Hilbert spaces. We recall
that a normed space X is called transitive if, given arbitrary norm-one
elements z, y E X , there exists a surjective linear isometry T : X + X such
that T ( z )= y. The notion of transitivity just introduced will be revisited
more quietly in Subsection 5.1.
Proof of Proposition 1.4. Let A be an absolute-valued algebra over K.
If A is a division algebra, then the normed space of A is transitive, since for
all norm-one elements x , y E A we have T ( z )= y, where T := L R G ~ ( is
vl
a surjective linear isometry on A . Therefore, when A is finite-dimensional,
Auerbach’s result applies, giving that the norm of A comes from an inner
product. Finally, if R = R,if A is finite-dimensional, and if A has a unit,
then A is equal t o either R,@, W,or 0 (by Hurwitz’s theorem). 0
The argument in the above proof is taken from page 156 of 6 , where
no reference to the works of Albert and Auerbach is done. In fact, Proposition 1.4 appears as Theorem 1 of 6 , and is directly attributed there to
Hurwitz 53, including shorty later the above argument as a part of the
complete proof of such Hurwitz’s theorem. We do not agree with this attribution. Indeed, as far as we know, the observation that absolute-valued
division algebras have transitive normed spaces appears first in the proof of
Lemma 4 of Wright’s paper log (fifty five years after Hurwitz’s paper). On
the other hand, Aurbach’s result, published thirty six years after Hurwitz’s
paper, seems to us non obvious.
2.3. The commutative Urbanik- Wright theorem
The second surprising result in the Urbanik-Wright paper
ing.
Theorem 2.3. R, C, and
real algebras.
lo6is
the follow-
are the unique absolute-valued commutative
We shall call Theorem 2.3 above the commutative Urbanik-Wright
theorem. We know no proof of Theorem 2.3 other than the original one
in lo6. Starting with a new application of Schoenberg’s theorem 95, such a
proof is really clever and easy. Therefore we do not resist the temptation of
reproducing it here. Some unnecessary complications are of course avoided.
113
Proof. Let A be an absolute-valued commutative real algebra. Since for
all norm-one elements z, y E A we have
4 = 411zYll = I(.
+ d2- .(
- Y)"I
i 112 + Y1I2
+ 11%
- YII
2
7
Schoenberg's theorem applies giving that A is a pre-Hilbert space. On the
*
other hand, since R, @, and @ are the unique absolute-valued commutative
real algebras of dimension 5 2 (see Subsection 1.4), it is enough to show
that the dimension of A is 5 2. Assume to the contrary that we can
find pair-wise orthogonal norm-one elements u,w,w in A. Then we have
11u2- v211 = IIu +v1111u - 1111 = 2. Since llu211 = 11v211 = 1, the parallelogram
law implies that u2+v2 = 0. Analogously, we obtain u2+w2 = v2+w2 = 0.
It follows u2 = 0, and hence also u = 0, a contradiction.
With the help of Lemma 2.4 below, the commutative Urbanik-Wright
theorem can be refined as follows. There is a universal constant K > 0 such
that every absolute-valued real algebra A satisfying llzy - yzll 5 Kllzllllyll
f o r all z, y
EA
is in fact equal to either R, @, or
(see Corollary 1.4 of
59).
Remark 2.1. For a normed space X over K, consider the property P which
follows:
( P ) There exists a normed space Y over R, together with a bilinear
mapping ( a , b ) + ab from X x X to Y satisfying ab = ba and
llabll = llall llbll for all a, b E X.
Arguing as in the beginning of the proof of Theorem 2.3, we see that, if the
normed space X satisfies Property P , then X is a pre-Hilbert space. The
converse is also true (see Theorem 4.4 of 8).
2.4. Power-associativity
Let A be an algebra over a field IF. We say that A is of bounded degree if
there exists a natural number n such that all single-generated subalgebras
of A have dimension 5 n, and power-associative if all single-generated
subalgebras of A are associative. In the case that the characteristic of IF is
different from 2, we will consider the algebra A" whose vector space is the
same as that of A , and whose product is defined by z @ y := $ ( ~ y yz).
We remark that both the bounded degree and the power-associativity pass
from A to A".
+
114
Lemma 2.1. Let A be a normed algebra over K satisfging llx211 = llx112 for
every x E A, and such that A" is power-associative and of bounded degree.
Then A has a norm-one unit.
Proof. Since A" is a commutative power-associative algebra of bounded
degree, and has no nonzero element x such that x 2 = 0, it follows from
Proposition 2 of 21 that A" has a unit element (say 1). Moreover, since
11111 = 111211 = 111112,we have 11111 = 1. Then, since A is a normed algebra,
both L1 and R1 lie in the closed unit ball of the normed algebra C ( A ) of
all continuous linear operators on A. Since i ( L 1 R1) = I A (the identity
operator on A ) , and I A is an extreme point of the closed unit ball of C ( A )
(by Proposition 1.6.6 of 9 3 ) , it follows that L1 = R1 = I A , i.e., 1 is a unit
element for A.
0
+
Now we can prove the main result in this subsection. It is due to ElMallah and Micali 45, and reads as follows.
Theorem 2.4. R, C,W, and 0 are the unique absolute-valued powerassociative real algebras.
Proof. Let A be an absolute-valued power-associative real algebra. By
Proposition 1.3, A is of bounded degree. Then, by Lemma 2.1, A has a
unit. Finally, by the noncommutative Urbanik-Wright theorem, A is equal
to either R, C,W,or 0.
0
The original proof of El-Mallah and Micali differs not too much of the
above one. Of course, they did not know Lemma 2.1, which has been proved
here by the first time. Thus, in the El-Mallah-Micali proof, Lemma 2.1
was replaced with a simpler purely algebraic result (see Lemma 1.1 of 45).
Anyway, both Lemma 1.1 of 45 and Proposition 2 of l7 (which has been
one of the tools in the proof of Lemma 2.1, and is also of a purely algebraic
nature) have a common root, namely the proof of Lemma 5.3 of 94. Before
the appearance of the Urbanik-Wright paper, Wright knew that R, C,W,
and 0 are the unique unital absolute-valued power-associative real algebras
(see the introduction of log). This (today doubly unsubstantial) result
was rediscovered by L. Ingelstam 54 (four years after the appearance of
the Urbanik-Wright paper!) with a proof essentially identical to the one
suggested by Wright in log. Anyway, the Wright-Ingelstam argument has
some methodological interest. Indeed, it shows that, in an autonomous
115
proof of Theorem 2.4, the noncommutative Urbanik-Wright theorem can
be replaced with Albert’s forerunner given by Proposition 1.6.
As we commented in Subsection 2.2, smooth normed algebras are preHilbert spaces. A converse to this fact is proved in Proposition 2.1 immedi(see
ately below. The key tools are Lemma 2.1 and the result of B. Zalar
also Theorem 3 of ‘12) that IR and C are the unique pre-Hilbert associative
commutative real algebras A satisfying llz211= )1z112f o r every z E A.
Proposition 2.1. Let A be a normed real algebra. Then the following
conditions are equivalent:
(1) A is a smooth-normed algebra.
(2) A is power-associative, the norm of A derives from a n inner product, and the equality 1)z211= )1z112holds f o r every z E A .
(5’) A” is power-associative, the norm of A derives from an inner product, and the equality llz211= ))z)12holds f o r every z E A .
Proof. The implication (1) + (2) is a consequence of Theorem 27 of 8 2 ,
whereas the one (2) + (3) is clear. Assume that Condition (3) is fulfilled.
Then, by Zalar’s result quoted above, the algebra As is of bounded degree.
Therefore, by Lemma 2.1, A has a norm-one unit. Since pre-Hilbert spaces
are smooth at all their norm-one elements, it follows that A is a smoothnormed algebra.
0
The following result of Zalar ‘11 follows straightforwardly from Proposition 2.1 above and Hurwitz’s theorem (see Subsection 2.2).
Theorem 2.5. Let A be a n absolute-valued real algebra whose norm springs
from a n inner product, and such that AS is power associative. Then A is
equal to either W,C, W,or 0.
In relation to Proposition 2.1, it is worth mentioning that smooth
normed algebras are precisely those unital normed algebras A satisfying
111 - z211 = 111 zlllll - zll f o r every z E A , as well as those unital
normed algebras A satisfying IIVz(y)II = 1 1 ~ ) ) ~ )f o) yr Jall
I z,y E A , where
U%(y):= z(yz) (yz)z - yz2 (see Corollary 29 of ”). Another characterization of smooth normed algebras is given in the next proposition.
+
+
Proposition 2.2. Let A be a normed real algebra. Then the following
conditions are equivalent:
(1) A is a smooth-normed algebra.
116
(2) A is power-associative, and the equality IIUz(y)II = llx11211~llholds
for all x, y E A.
Proof. In view of Proposition 2.1 and the comments immediately above,
it is enough t o show that (2) implies that A has a unit. Assume that (2) is
fulfilled. Then for x, y in any single-generated subalgebra of A, we have
Therefore, all single-generated subalgebras of A are absolute-valued algebras. By Proposition 1.3, A is of bounded degree. Finally, by Lemma 2.1,
A has a unit.
0
Proposition 2.2 was first proved by M. Benslimane and N. Merrachi lo
with slightly different techniques. More information about smooth normed
algebras can be found in Subsection 3.5.
To conclude the present subsection, let us comment that Theorem 2.4 is
“almost” contained in the early paper of Urbanik lo2. Indeed, it could have
been very easy for him to establish such a theorem by selecting, among
the many auxiliary results in that paper, the appropriate ones for the goal.
However, Urbanik does not do this, since he completely devotes his paper
lo2 to characterize R, @, IHI, and 0 in terms conceptually far from the
power-associativity. An element x of an algebra A is said to be reversible
if there exists y E A satisfying x y - xy = x y - yx = 0. The algebra A is
said to fulfill the reversibility condition if all its elements, except those
in some countable set, are reversible. Now the main result in lo2 reads as
follows.
+
+
Theorem 2.6. R, @, W,and 0 are the unique absolute-valued real algebras
satisfying the reversibility condition.
Note that for A equal to either R, @, IHI, or
the unit of A, are reversible.
0,
all elements of A, except
2.5. Flexibility
An algebra is said to be flexible whenever it satisfies the identity
( 2 1 2 2 ) q = x 1 ( z 2 q ) . Since single-generated subalgebras of flexible algebras are commutative, the commutative Urbanik-Wright theorem applies
successfully t o single-generated subalgebras of absolute-valued flexible algebras. After a lot of work, involving the information obtained from the
procedure just pointed out, El-Mallah and MiCali 46 prove the following.
117
Lemma 2.2. Absolute-valued flexible algebras are finite-dimensional.
Later, El-Mallah, in a series of papers (see 36, 37, 38, 39, and 40), refines
deeply the result just reviewed, by considering absolute-valued algebras satisfying the identity xx2 = x2x (which is of course implied by the flexibility),
and proving the following.
Theorem 2.7. For a n absolute-valued real algebra A, the following assertions are equivalent:
(1) A is flexible.
(2) A i s a pre-Hilbert space and satisfies the identity x2x = x x 2 .
(3) A i s finite-dimensional and satisfies the identity x 2 x = xx2.
*
*
*
(4) A i s equal to either R, C, C, Ell, H, 0, 0, or the algebra
pseudo-octonions.
P of
According to Theorem 2.7 just formulated, the algebra P of pseudooctonions is the unique absolute-valued flexible real algebra which has
been not still introduced in our development. Such an algebra was discovered by S. Okubo 72 (see also pages 65-71 of 70). The vector space of P is
the eight-dimensional real subspace of M3 (C) consisting of those trace-zero
elements which remain fixed after taking conjugates of their entries and
passing t o the transpose matrix. The product 0 of P is defined by choosing
a complex number p satisfying 3p(1 - p ) = 1, and then by putting
1
x 0y := p ~ y (1 - / L ) ~-xz T ( x y ) l .
+
Here T denotes the trace function on M3(C), 1 stands for the unit of the
associative algebra I@(@),
and, for x , y in P,x y means the product of x and
y as elements of such an algebra. If for x , y E P we define ( x l y ) := i T ( x y ) ,
then (.I.) becomes an inner product on P whose associated norm is an
absolute value.
In relation to Theorem 2.7, it seems t o be an open problem (see the
abstract of 41) if every absolute-valued real algebra satisfying the identity
x2x = xx2 is finite-dimensional. According to Theorem 2.7 itself, the answer is affirmative if A is a pre-Hilbert space. The answer is also affirmative
if A is algebraic 41, but, as we will see in Subsection 2.7, this result is today
unsubstantial. As a more ambitious problem, we can wonder whether every
absolute-valued algebra satisfying some identity is finite-dimensional.
The classification of absolute-valued flexible real algebras contained in
Theorem 2.7 was tried in 63, with a partial success. Actually, such a clas-
118
sification can be derived from Lemma 5.3, Proposition 1.5, and 73. Theorem 2.7 has inspired the result in 34 that finite-dimensional composition
algebras satisfying the identity x 2 x = xx2 are an fact flexible.
2.6. H*-theory
The following theorem has been proved by J. A. Cuenca and the author
26.
Theorem 2.8. Let A be an absolute-valued algebra over K. Assume that
there exists a complete inner product (.I.) o n A, together with an involutive
conjugate-linear operator * o n A, satisfying (zylz) = ( x l z y * ) = (y(x*z)f o r
all x,y , z E A . Then we have:
(1) A is finite-dimensional.
(2) The Hilbertian norm. x +
is a positive multiple of the
absolute-value of A.
(3) The operator * is an algebra involution o n A .
(4) The equality x*(zy) = (yx)x* =
holds f o r all x,y, z E A.
With the terminology of 2 5 , the assumptions on (.I.) and * in Theorem 2.8 mean that, forgetting the absolute value of A, (A,(.[.),*)
is a
semi-H*-algebra over K, The conclusion, that * is in fact an algebra involution, then reads as that ( A ,(.I.), *) is an H*-algebra over K. Besides
a little H*-theory 2 5 , the proof of Theorem 2.8 involves some results on
absolute-valued algebras previously reviewed (as Wright’s Theorem 1.3),
and others to be reviewed later (as for example Theorem 3.8). Such a
proof, as well as that of Theorem 2.9 below, also includes some easy facts
first pointed out in 86. Among these, we emphasize the following one for
later reference.
L e m m a 2.3. Let A be an absolute-valued algebra over K. Assume that the
absolute-value of A comes from an inner product (.I.), and that, for every
x E A, there exists x* E A satisfying (xylz) = (ylz*z) for all x,y, z E A.
f o r all x,y , z E A.
Then we have x * ( x y ) =
Proof. For x,y E A, we have (zylxy) = Ilxl12(yly). Linearizing in the
variable y, we obtain that the equality ( z z l x y ) = IIx112(zly) holds for all
x,y,z E A . Since ( x z I x y ) = (zlx*(xy)), we deduce ( z I z * ( z y ) )= 11x112(z1y),
which, in view of the arbitrariness of z , yields z * ( x y ) = l l ~ 1 1 ~ y .
0
The Cuenca-Rodriguez paper 26 also contains a precise determination
of the algebras A in Theorem 2.8. Since the case that K = C is unsub-
119
stantial (see Subsection 2.8 later), only the case that R = R merits to be
considered. Thus, in view of Theorem 2.8, we are dealing in fact with an
absolute-valued finite-dimensional real algebra A endowed with an algebra
involution *, and whose norm derives from an inner product (.I.) satisfying
x*(xy) = (yx)x* and (xylz) = (xlzy*) = (ylx*z) for all z,y,z E A. Since
* is isometric, we can consider the isotope of A (say B ) consisting of
the normed space of A and the product x @ y := x*y*. Now, we trivially realize that the absolute-valued real algebra B is flexible and satisfies
(x0ylz) = (zly 0z ) for all z, y, z E B . Then, we deduce from El-Mallah’s
* * *
Theorem 2.7 that B is equal to either R, @, W,0, or P. Moreover, * becomes an algebra involution on B , and the correspondence (A,*) 4 ( B ,*)
is categorical and bijective. After the laborious classification of algebra in-
* * *
volutions on @, W,0, and P made in 26, the determination of the algebras
in Theorem 2.8 concludes. In this way, three new distinguished examples
of absolute-valued finite-dimensional real algebras appear. These are the
natural isotopes of MI,0, and P (denoted respectively by W,0, and @ built
as follows. For every absolute-valued algebra A, and every linear isometry
$ on A, the $-twist of A is defined as the absolute-valued algebra consisting of the normed space of A and the product x @ y := $(x)$(y). For A
equal to either JH[ or 0,
we define as the $-twist of A, where $ stands
for the essentially unique involutive automorphism of A different from the
identity operator (see Subsection 1.4). On the other hand, there exists an
“essentially” unique algebra involution 0 on P, which allows us to define @
as the 0-twist of P. Now we have the following.
A
h
Theorem 2.9. Let A be an absolute-valued real algebra fulfilling the requirements in Theorem 2.8. Then A is equal to either R,@, ,; JHI, fi, 0,
6,o r @ .
A slight variant of the proof of Theorem 2.9 sketched above, involving
Corollary 7 of 74 instead of Theorem 2.7, can be seen in Remark 2.9 of 26.
2.7. Alge braicit y
Albert’s Proposition 1.6, although obsolete after the noncommutative
Urbanik-Wright theorem, has had the merit of encouraging the work on
the question if every absolute-valued algebraic algebra is finite-dimensional.
Since for complex algebras such a question has an almost trivial affirmative
answer (see the concluding paragraph of Subsection 2.8 below), the interest
centers in the case of real algebras. Some partial affirmative answers have
120
been provided by El-Mallah. Thus, an absolute-valued algebraic real algebra is finite-dimensional whenever there exists a nonzero idempotent in A
commuting with every element of A 36, or there exists a continuous algebra
involution * on A satisfying xx* = x*x f o r every x E A 3 9 J or A satisfies
the identity xx2 = x2x 41. We note that the result in 36 would become later
a consequence of the one in 39 (see El-Mallah's Theorem 3.2), and that the
result in 41 was already commented a t the conclusion of Subsection 2.5.
To specify that an algebra A is of bounded degree, let us say that A
is of degree n E N if n is the minimum natural number such that all
single-generated subalgebras of A have dimension 5 n. It follows from
Proposition 1.4, that absolute-valued algebraic real algebras are of bounded
degree, and, more precisely, of degree 1,2,4, or 8. Then, since IR is the
unique absolute-valued algebraic algebra of degree 1 (see again the concluding paragraph of Subsection 2.8), the strategy of studying separately the
cases of degree 2,4, and 8 could seem tempting in order t o answer affirmatively the question we are considering. Unfortunately, such an strategy
has turned out t o be unsuccessful for the moment, unless for the case of
degree 2, for which we have the following result of the author 89.
*
Theorem 2.10. The absolute-valued real algebras of degree two are @, @,
h, *HIJ MI*, 0,6,*0,O*, and IF'.
*@, @*, W,
Via the commutative Urbanik-Wright theorem, Theorem 2.10 above
contains both Theorem 2.4 and the classification of absolute-valued flexible
real algebras included in Theorem 2.7. However, this is quite deceptive
because, in fact, the proof of Theorem 2.10 involves Theorem 2.4 and the
whole Theorem 2.7. In any case, by keeping in mind again the commutative
Urbanik-Wright theorem, Theorem 2.10 shows by the first time that, for
absolute-valued algebras, power-commutativity and flexibility are equivalent
notions. We recall that an algebra is said to be power-commutative if all
its single-generated subalgebras are commutative, and that flexible algebras
are power-commutative 7 6 . Theorem 2.10 has inspired the classification of
composition algebras of degree two, done in 34.
Returning to the general problem if absolute-valued algebraic algebras
are finite-dimensional, we must say that, six years ago, A. Kaidi, M. I.
Ramirez, and the author 57 succeeded in solving it. Thus we have the
following.
Theorem 2.11. A n absolute-valued real algebra is finite-dimensional if
(and only i f ) it is algebraic.
121
We know no proofs of Theorem 2.11 above others than the original one
in 57, and the slight variant of it given in 58. We do not enter here the details
of such proofs, nor even give a sketch of them. Referring the reader to 58 for
such a sketch, we limit ourselves here to say that both arguments are long
and complicated, and involve in an essential way the techniques of normed
ultraproducts 52. Thus, by the first time in the theory of absolute-valued
algebras, the following folklore result shows useful.
Lemma 2.4. The normed ultraproduct of every ultrafiltered family of
absolute-valued algebras over R becomes naturally an absolute-valued algebra over R.
Concerning the proof of Theorem 2.11, let us also revisit a minor auxiliary result (namely, Lemma 4.2 of 5 7 ) . Such a result can be refined as
follows.
Lemma 2.5. Let X be a normed space over R, let F : X -+ X be a linear
contraction, and let M be a finite-dimensional subspace of X . Assume that
F is the identity o n M , and that X is smooth at every norm-one element
of M . Then there exists a continuous linear projection IT from X onto M
such that ker(7r) is invariant under F .
Proof. Let M * denote the dual space of M . By a theorem of Auerbach
(see also Lemmas 7.1.6 and 7.1.7 of 92), there are bases {ml,...,m k } and
(91, ...,gk} of M and M * , respectively, consisting of norm-one elements and
satisfying gi ( m j )= S i j . Extending each gi to a norm-one linear functional
& on X (via the Hahn-Banach theorem), and considering the mapping
2 + ~ ~ = , q 5 i ( z )from
m i X to M , it is easily seen that such a mapping
satisfies the properties asserted for 7r in the statement of the lemma (see
the proof of Lemma 4.2 of 57 for details).
0
Lemma 2.5 above was proved in 57 under the additional assumption
that the restriction to M of the norm of X springs from an inner product.
The refinement we have just made does not matter there because, when
the lemma applies, X is an absolute-valued algebra, and M is a subspace
of a finite-dimensional subalgebra of X , so that the superfluous requirement in the original formulation of the lemma is automatically fulfilled (by
Proposition 1.5).
122
2.8. A remark on complex algebras
All conditions we have considered above, leading absolute-valued real algebras to the finite-dimension, in the case of absolute-valued complex algebras yield that the algebra is C. Indeed, if an absolute-valued complex
algebra fulfills some of those conditions, then, by restriction of scalars, we
obtain an absolute-valued real algebra satisfying the same condition, and
hence the corresponding result applies. But we know that absolute-valued
finite-dimensional algebras are division algebras, and that C is the unique
finite-dimensional division complex algebra.
In some cases, the result obtained in this way can be refined still more.
For example, the complex version of Theorem 2.2 is that, i f A is an absolutevalued complex algebra, and i f there exists a E A such that a A is dense in
A, then A = C (see Lemma 1.1of 2 6 ) . On the other hand, the joint complex
version of Theorems 2.8 and 2.9 is that, if A is an absolute-valued complex
algebra, and i f there exists a complete inner product (.I.) on A making
the product continuous, and an involutive conjugate-linear operator * o n A
satisfying (zylz) = (zlzy*) for all x , y , z E A, then A = C (see Theorem
1.2 of ")). None of the two results just quoted remains true (with the
finite-dimensionality of A instead of A = C in the conclusion) whenever
real algebras replace complex ones. Concerning the second result, in the
real case nor even can be expected the Hilbertian norm z +
to
be equivalent to the absolute value of A (see Example 1.7 of 2 6 ) . These
pathologies give rise to an interesting development of the theory of absolutevalued algebras, which will be reviewed in Subsection 3.5.
As a consequence of Theorem 2.11 and the comments at the beginning of
the present subsection, C is the unique absolute-valued algebraic complex
algebra. However, this can be proved elementarily. Indeed, notice that,
by the same comments, absolute-valued algebraic complex algebras are of
degree one, and that, if F is a field containing more than two elements, if
A is an algebra over F of degree one, and if there is no nonzero element
x E A with x2 = 0, then A = F (see for example page 297 of 57).
3. Infinite-dimensional absolute-valued algebras
3.1. The basic examples
The first example of an absolute-valued infinite-dimensional algebra appears in the celebrated paper of Urbanik and Wright lo6. Indeed, they
show that the classical real Hilbert space l 2 becomes an absolute-valued
algebra under a suitable product. Looking at their argument, many other
123
similar examples can be built. To get them, let us start by fixing an arbitrary nonempty set U , and a mapping 6 : U x U --f X , where X = X ( U , K )
stands for the free vector space over K on U . We denote by A = A(U,6,K)
the algebra over K whose vector space is X , and whose product is defined as
the unique bilinear mapping from X x X to X which extends 6. From now
on, we assume that U is infinite, and, accordingly to such an assumption,
we choose 6 among the injective mappings from U x U to U or, more generally, of the form f g , where g : U x U ---t U is injective and f : U x U + K
satisfies If(u,v)l = 1 for every ( u , v ) E U x U . With these restrictions
in mind, we are going to realize that there are “many” absolute values on
A. To this end, let us involve a new ingredient, namely an extended real
number p with 1 5 p 5 00. Then, for x in A, we can think about the family
{x,},~u of coordinates of x relative to U ,and define
llzllp := (CuEu
Iz,IP)t if p
< 00 and IIxllm := max{1z,I
:uE
U}.
Invoking the properties of 6, we straightforwardly verify that )I . 1, is an
absolute value on A. We denote by dp= dp(U,19, K)the absolute-valued
algebra over K obtained by endowing A with the norm 11. .,1 By considering
the completion of A,, we obtain a complete absolutevalued algebra over R,
denoted by C, = C,(U, 6, K), whose Banach space is nothing other than the
familiar space l,(U, K) if p # 00,or Q(U,K)otherwise. Now, the UrbanikWright example is just the algebra Cz(N,6, R), with 6 : N x N --f N equal
to any bijection.
Returning to our general setting, let us remark that, since C, is a Hilbert
space if and only if p = 2, it follows from the above construction that composition algebras need not be finite-dimensional, and that, contrarily to
what is conjectured in 40, absolute-values need not come from inner products. Another consequence of our construction is that there exist complete
absolute-valued algebras without uniqueness of the (noncomplete) absolute
value. Indeed, for 1 5 p < q 5 00, the complete absolute-valued algebra
C, can be algebraically regarded as a subalgebra of C,, but the topology of
the restriction of the absolute value of C, t o C, does not coincide with the
natural one of C,. The straightforward fact, that 11 1, 2 11 . 1, on C,, is
not anecdotic. Indeed, as a consequence of Theorem 3.8 below, every complete algebra norm o n an absolute-valued algebra is greater than the absolute
value. In particular, two complete absolute values on the same algebra must
coincide.
The refinement of the Urbanik-Wright example, done above, is implicitly known in some works on Banach spaces (see for instance the proof of
-
124
Theorem 3.a.10 of 6 5 ) . The interest of such a refinement in the theory of
absolute-valued algebras seems to have been first pointed out in 85.
A real algebra A is said to be ordered if it is provided with a subset
A+ of positive elements, which is closed with respect t o multiplication by
positive real numbers and with respect to addition and multiplication in A ,
and satisfies A+ n ( - A + ) = 0 and A+ U ( -Af) = A \ (0). In lo4,Urbanik
shows that IR is the unique absolute-valued finite-dimensional ordered real
algebra. Nevertheless, he also proves the following.
Theorem 3.1. There exists a com.plete absolute-valued infinite-dimensional ordered real algebra.
A simplification of Urbanik’s argument is the following.
Proof. Let 6 : N x N + N be defined by 6(n,m):= 2n3m, and let us fix
1 5 p 5 00. Since 6 is injective, we can consider the complete absolutevalued infinite-dimensional real algebra C, = C,(N, 6,R). The natural inclusion N -+ C, converts N into a Schauder basis of C,. For z E C,, let
{ z n } , E ~ stand for the family of coordinates of z relative t o such a basis,
and, when z # 0, define n ( x ) := min{n E N : IC, # 0). Finally, put
C t := {x E C, \ (0) : zrn(%)
> 0). Keeping in mind that 19 is increasing
in each one of its variables, it is easily seen that C$ fulfils the properties
required above for the sets of positive elements of ordered real algebras. 0
3.2. h e normed nonassociative algebras
Let us fix a nonempty set V . Nonassociative words with characters in
V are defined inductively (according t o their “degree”) as follows. The
nonassociative words of degree 1 are just the elements of V . If n 2 2, and
if we know all nonassociative words of degree < n , then the nonassociative
words of degree n are defined as those of the form ( W I ) ( W ~ )where
,
w1 and
w2 are nonassociative words with deg(wl)+deg(w2) = n. Although the use
of brackets is essential in the above definition, some natural simplifications
in the writing are usually accepted. For example, brackets covering a word
of degree 1 are omitted, and words of the form (w)(w), for some other
word w ,are written as ( w ) ~ Two
.
nonassociative words are taken to be
equal only if they have exactly the same writing. Thus for example, for
v E V , the nonassociative words vv2 and v2v are different. Now, denoting
by U the set of all nonassociative words with characters in V , and by 6 the
mapping ( ~ 1 , 2 0 2-+
) (w1)(w2) from U x U to U , we can think about the
125
algebra A(U, 6, K)constructed in the preceding subsection. Since such an
algebra depends only on V and K,we denote it by F(V,K). The algebra
.F(YiK)] called the free nonassociative algebra on V over K,contains
V in a natural manner, and is characterized up to algebra isomorphisms
by the following “universal property”: If A is any algebra over K,and
if cp : V + A is any mapping, then ‘p extends uniquely to an algebra
homomorphism from F(V,K) to A (see Theorem 1.1.1 of ‘13). Now, since
the mapping 29 above is injective (by Proposition 1.1.2 of ‘13), we invoke
again the preceding subsection to realize that there are “many” absolutevalues o n F(V,IK). In the original proof Io4 of Theorem 3.1, Urbanik already
knows that, when V reduces to a singleton, F(V,R) becomes an absolutevalued algebra under the norm 1). 112. The general case of such an observation
is due to M. Cabrera and the author (who announced it in 16), and appears
formulated with the appropriate precisions first in 8 5 . For 1 5 p 5 co,
we denote by Fp(V,IK)the absolute-valued algebra over IK obtained by
endowing F(V,
K) (= d(U,6,K)for U and 29 as above) with the absolute
value 11 . I l p . As we are seeing in the proof of Proposition 3.1 immediately
below, the absolute-valued algebra F1 (V, K)has a special relevance in the
general theory of normed algebras.
Proposition 3.1. Let V be a nonempty set. Then, up to isometric algebra
isomorphisms, there exists a unique normed algebra N = N ( V ,K) over K
satisfying the following properties:
( 1 ) V is a subset of the closed unit ball of N.
(2) If A is any normed algebra over IK, and if cp is any mapping from
V into the closed unit ball of A , then cp extends uniquely to a contractive algebra homomorphism from N to A .
Moreover, we have:
(3) The normed algebra N is in fact an absolute-valued algebra.
(4) The set V consists only of norm-one elements of N .
Proof. Take N = Fl(V,K). Clearly N satisfies Properties ( I ) , (3), and
(4) in the statement. Let A be a normed algebra over IK, and let cp be a
mapping from V into the closed unit ball of A. Since, forgetting the norm,
N is nothing other than F(V,IK), the universal property of this last algebra
provided us with a unique algebra homomorphism 11, : 3
1(V,K)--f A which
extends cp. Let x be in N . We have x = CwEU
x w w , where U denotes the
set of all nonassociative words with characters in V , and { I C , } , ~ ~stands
126
for the family of coordinates of x relative to
U.Therefore
(Starting from the fact that $ ( V ) is contained in the closed unit ball of
A , the inequality ll$(w)II 5 1 just applied is proved by induction on the
degree of w.) Now that we know that N also satisfies Property (2), let us
conclude the proof by showing that Af is the “unique” normed algebra over
K satisfying (1) and (2). Let N’be a normed algebra over K satisfying
(1) and (2) with N ’instead of N , Then we are provided with contractive
algebra homomorphisms 4 : N -+ N ’and 4’ : N ’+ N fixing the elements
of V . Therefore $04 and 40# are contractive algebra endomorphisms of N
and N ‘, respectively, extending the corresponding inclusions V 4 N and
V -+ N I.By the uniqueness of such extensions, we must have 4’ 0 4 = IN
and c$ o 4’ = I,v It follows that @ is an isometric algebra isomorphism
from N onto N ’respecting the corresponding inclusions of V in each of
0
the algebras.
I .
Now, if A is a normed algebra over K, if V denotes the closed unit ball of
A , and if CP : N(V,K) -+ A is the unique contractive algebra homomorphism
which is the identity on V, then we easily realize that the induced algebra
homomorphism N(V,K)/ ker(@) A is a surjective isometry. Therefore,
we have the following.
-+
Corollary 3.1. Every normed algebra over K is isometrically algebraisomorphic to a quotient of an absolute-valued algebra over K.
The absolute-valued algebra N(V,K) in Proposition 3.1 has its own
right to be called the free normed nonassociative algebra on the set
V over K. The variant of Proposition 3.1, with Lccompletenormed” instead
of “normed” everywhere, is also true, giving rise to the free complete
normed nonassociative algebra on the set V over K. This algebra is
implicitly involved in the proof of the following result.
Corollary 3.2. Every complete normed algebra over K is isometrically
algebra-isomorphic to a quotient of a complete absolute-valued algebra
over K.
Proof. Let A be a complete normed algebra over K. Choose any subset
V of A whose closed absolutely convex hull is the closed unit ball of A .
By Proposition 3.1, N(V,K) is an absolute-valued algebra over K whose
127
closed unit ball contains V , and there exists a contractive algebra homomorphism from N(V,IK)to A fixing the elements of V . By passing to the
completion of N(V,IK), and invoking the completeness of A, we are in fact
provided with a complete absolute-valued algebra B over IK whose closed
unit ball contains V , and a contractive algebra homomorphism @ : B
A
fixing the elements of V . Let A1 and B1 denote the closed unit balls of A
and B , respectively. Since a ( & ) is an absolutely convex subset of A containing V, and A1 is the closed absolutely convex hull of V, the closure of
@(B1)in A contains A l . Now, from the main tool in the proof of Banach’s
open mapping theorem (see for example Lemma 48.3 of 11) we deduce that
@(B1)contains the open unit ball of A. Since
: B -+ A is a contractive
algebra homomorphism, it follows from the above that the induced algebra
homomorphism B / ker(@) 4 A is a surjective isometry.
0
-+
Of course, the most confortable choice of V in the above proof is the
one V = A1 . However, finer selections of V allow us to realize that the
absolute-valued algebra B can be chosen with the same density character
as that of A. We recall that the density character of a topological space
E is the smallest cardinal among those of dense subsets of E.
Gelfand-Naimark algebras are defined as those complete normed
complex algebras A endowed with a conjugate-linear algebra involution *
satisfying 11x*x11 = 1 1 ~ 1 for
1 ~ every x E A. Their name is due to the celebrated Gelfand-Naimark theorem 30 that there are no Gelfand-Naimark
associative algebras others than the closed *-invariant subalgebras of the
Banach algebra C ( H ) of all continuous linear operators o n some complex
Hilbert space H , when this last algebra is endowed with the involution * determined by (x(q)I<)= (qlx*(<))for every x E C ( H ) and all q,< E H . The
nonassociative Gelfand-Naimark theorem
asserts that unital GelfandNaimark algebras are alternative. Moreover, every alternative GelfandNaimark algebra can be seen as a closed *-invariant subalgebra of a unital
Gelfand-Naimark algebra, and the study of alternative Gelfand-Naimark
algebras can be reasonably reduced to that of associative ones and to that
of the complexification of 0 with suitable norm and involution. For these
and other interesting results in the theory of Gelfand-Naimark alternative
algebras the reader is referred to 56 and references therein. Now, absolutevalued algebras provide us with examples of Gelfand-Naimark algebras
which are not alternative. Indeed, it follows easily from Proposition 3.1
that, for any nonempty set V , the absolute-valued algebra N(V,C ) has an
isometric conjugate-linear algebra involution fixing the elements of V . By
128
passing to the completion, we obtain an absolute-valued Gelfand-Naimark
algebra which is not alternative (nor even satisfies any identity when V
is infinite). As pointed out in 8 7 , the same remains true if we start from
Fp(V,C)(1 5 p I
m) instead of N(V,C) (= .?’l(V,C)).
3.3. Center, centroid, and extended centroid
Let A be an algebra over a field IF. For x, y, z E A, we write [z,y] := xy- yx
and [x,y, z ] := (xy)z-z(yz). The center of A (denoted by Z ( A ) )is defined
as the set of those elements x E A such that
and is indeed an associative and commutative subalgebra of A. The centroid of A (denoted by r ( A ) )is defined as the set of those linear operators
f on A satisfying f ( z y ) = f ( z ) y = xf(y) for all x , y E A , and becomes
naturally an associative algebra over IF with a unit. Under the quite weak
assumption that there is no nonzero element x E A with XA = Ax = 0,
the associative algebra r ( A ) is also commutative, and, by identifying each
element z E Z(A) with the operator of left multiplication by z on A, Z ( A )
imbeds naturally into r ( A ) . From now on, assume that A is prime (i.e.,
PQ # 0 whenever P and Q are nonzero (two-sided) ideals of A ) . Then
r ( A ) becomes an integral domain, and hence it can be enlarged to its field
of fractions. However, such an enlargement does not provide any additional
information on the structure of A. By the contrary, a larger field extension of r ( A ) , called the extended centroid of A and denoted by C ( A ) ,
has turned out to be very useful to determine the behaviour of A 47. The
elements of C ( A ) are those linear mappings f : Pf
A, where Pf is some
nonzero ideal of A, satisfying f(xp) = xf(p) and f ( p z )= f(p)x for every
(z, p ) E A x P f . Two elements f , g E C ( A ) are considered to be “equal” if
they coincide on Pj n Pg. Summing and composing elements of C ( A ) as is
usually done for partially defined operators, such sum and composition are
compatible with the notion of “equality” settled above, and convert C ( A )
into a field extension of IF. Moreover, r ( A ) imbeds naturally into C ( A ) .
Now, if A is an absolute-valued real algebra, then, by Theorem 2.1, we
have Z ( A ) = 0 unless A is equal to either R,C, IHI, or 0.As a consequence, Z ( A ) = 0 for every absolute-valued complex algebra A different
f r o m C. Noticing that every absolute-valued algebra A is a prime algebra,
the determination of r ( A ) follows from the inclusion r ( A ) C C ( A ) , and
Proposition 3.2 immediately below.
--f
129
Proposition 3.2. Let A be an absolute-valued algebra over R. Then
C ( A )= @ if K = C , and C ( A ) is equal to either R or C if K = R.
Proof. In view of Lemma 1.1, it is enough to show that C ( A ) can be
endowed with an absolute value. To this end, we claim that, if f,g are in
C ( A ) ,if f is “equal” to g, and if p and q are norm-one elements of Pf and
P,, respectively, then 11 f ( p ) ” = 11g(q)11. Indeed, p q lies in Pf n P,, so we
have f ( p ) q = f (P4) = d p q ) = p g ( q ) , and hence
Ilf(P)ll = Ilf(P)nll = Ilpg(q)ll = 119(q)11~
as desired. Now f 4 llf(p)II, with f and p as above, becomes a (welldefined) real valued mapping on C ( A ) , and it is easily seen that such a
0
mapping is an absolute value.
Proposition 3.2 above can be derived either from Theorem 3 and Remark
2 of l6 (by keeping in mind Lemma 2.4), or straightforwardly from Corollary
1 of ”. The autonomous proof given here is taken from 86.
As a consequence of Proposition 3.2, if A is an absolute-valued algebra over K,then r ( A ) = C if K = @, and r ( A ) is equal to either R or
C if K = R. Let A be an absolute-valued real algebra. We can have
either C ( A ) = r ( A ) = R (as happens in the case A = R,W, or O),
C ( A ) = r ( A ) = @. (which happens if and only if A is the absolute-valued
real algebra underlying a complex one), or C ( A ) = C and r ( A ) = R. To
exemplify the last possibility, note that it is easily deduced from Proposition 3.1 the existence of a complete absolute-valued complex algebra B ,
together with a continuous nonzero algebra homomorphism 4 from B to C.
Taking v E B with 4(v) = 1, and putting A := Rv @ ker(4), A becomes
a closed real subalgebra of B (and hence, a complete absolute-valued real
algebra) such that C ( A ) = C and r ( A ) = R. Although Proposition 3.1 was
not explicitly known in 86, the example just reviewed appears there with
an argument essentially equal to that we have given here.
3.4. Algebras with involution
The following result is due to Urbanik
lol.
Proposition 3.3. Let A be an absolute-valued real algebra endowed with
an isometric algebra involution * which is different f r o m the identity operator and satisfies xx* = x*x f o r every x E A . Then self-adjoint elements
commute with skew elements, and there exists an idempotent e E A such
130
that the equality x*x = 11x112e holds for every x E A . As a Consequence, the
absolute value of A comes from an inner product.
Looking at B. Gleichgewicht’s paper 49, we discovered that the first assertion in the conclusion of Proposition 3.3 is nothing other than a joint
reformulation of Lemmas 1, 2, and 3 of lol. Keeping in mind such a reformulation, the consequence that A is a pre-Hilbert space, proved in Lemma
4 of l o l lseems t o us obvious.
Seventeen years after the appearance of Urbanik’s paper loll ElMallah 39 shows that, i f A is an absolute-valued real algebra fulfilling the
requirements in Proposition 3.3, then the comrnutant of e in A (say B ) is
in fact a self-adjoint subalgebra of A , and we have B = Re @ A,, where A ,
stands for the space of all skew elements of A . Shorty later, he proves the
remarkable converse which follows.
Theorem 3.2. 40 Let A be an absolute-valued real algebra containing a
nonzero idempotent e which commutes with all elements of A . Then the
absolute-value of A derives from an inner product (.I.). Moreover, the isometric mapping x 4 x* := 2 ( x J e ) e- x becomes an algebra involution o n A
satisfying x*x = xx* for every x E A .
The conclusion in Theorem 3.2, that A is a pre-Hilbert space, remains
true if the requirement of the existence of a nonzero idempotent which
commutes with all elements of A is relaxed to that of the existence of a
nonzero element a which commutes with all elements of A and satisfies
a(aa2) = ( a 2 ) 2 (see 42), El-Mallah’s paper 39, already quoted, contains
results non previously reviewed, some of which merits a methodological
comment. For instance, the proof of Theorem 5.6 of 39 (asserting that an
absolute-valued algebra A is finite-dimensional whenever so is the subspace
of A spanned by squares and there exists a E A \ (0) satisfying a x = xu for
every x E A ) can be concluded after its two first lines. Indeed, we have the
following.
Lemma 3.1. Let A be an absolute-valued algebra over K such that there
exists a E A \ (0) satisfying a x = x u for every x E A . Then A imbeds
linearly and isometrically into the subspace S ( A ) of A spanned by squares.
+
Proof. We may assume llull = 1. Since for x E A we have ( a x ) =
~
a2 2ax x2, we deduce L a ( A ) C_ S ( A ) . But La is a linear isometry.
+
+
131
Now, let us return to Urbanik's paper lo' to review its main results.
These are a construction method producing in abundance absolute-valued
real algebras A fulfilling the requirements of Proposition 3.3, and a theorem
characterizing the algebras obtained from such a construction. The ingredients of the construction are an infinite set U , a nonempty subset T of U
such that #(U \ 5") = #U (where # means cardinal number), an element
uo E T , an injective function 4 from the family of all binary subsets of U
to U whose range does not intersect T , and a function $ : U x U + (1, -1)
satisfying+(u,w)+$(v,u) = Owhenever ( u , v )E (TxT)U((U\T)x(U\T)),
and $(u,
u ) = 1 otherwise. Now, putting ~ ( u:=
)f l depending on whether
or not u belongs to T , and defining 19 : U x U -+ X ( U ,R) by
8(u,
w) := $(u,w)$({u,v}) if u # Y and 6(u,
u):= E ( U ) U O ,
we consider the associated real algebra A = A(U,d,R) in the meaning
of Subsection 3.1. After a careful calculation, we realize that A becomes
an absolute-valued algebra under the norm llxJJ:=
1x,)2)i, where
{ x , } , ~ ~is the family of coordinates of z relative to U . Moreover, the
unique linear operator * on A which extends the mapping u 4 E ( U ) U from
U to A becomes an isometric algebra involution satisfying x*x = xx* for
every x E A. If in addition we put ((sly)) := $(xy*
yx*), then we
have ((xylzt)) = ((xz*Iy*t)) for all z,y,z,t E A. Passing to the completion of A, we obtain a complete absolute-valued real algebra, denoted by
R = R(U,T ,U O ,4, +), which is endowed with an isometric algebra involution * satisfying x*x = xz* and ((xylzt)) = ((xy*lz*t))for all x , y , z , t E R,
where ((xly)) := :(xy*
yz*). Following lol, we codify the information
on R just collected by saying that R is a regular absolute-valued *algebra.
To classify regular absolute-valued *-algebras, Urbanik introduces a particular appropriate type of isotopy, called similarity. If A is a regular
absolute-valued *-algebra, and if F : A 4 A is a surjective linear isometry commuting with *, then the Banach space of A with the same involution becomes a new regular absolute-valued *-algebra under the product z @ y := F(xy). Algebras obtained from A by the above procedure
are called similar to A . By the way, two algebras R(U,T,UO,C$,+)
and
R(U',
TI,ub,4', $ I ) are similar if and only if #U = #U', #T = #TI, and
# S = #S', where S stands for the set of those elements of U which are
neither in T nor in the range of 4. Thus, each similarity class of the algebras in Urbanik's construction depends only on three cardinal numbers
W I , W ~ , Wwith
~
a 1 infinite, a 3 5 wl, and 0 # w2 I wl. Denoting by
(xuEU
+
+
132
such a similarity class, Urbanik’s structure theorem for regular absolute-valued *-algebras reads as follows.
~ ( w lw2,
, w3)
Theorem 3.3. Every regular absolute-valued *-algebra is similar to either
R, @ (with * equal to either the identity or the standard involution), or
one in the class X(w1, w2, w3) for suitable cardinal numbers w l , w2, w3 as
above.
Let A be an absolute-valued real algebra endowed with an isometric
algebra involution * which is different from the identity operator and satisfies xx* = x*x for every x E A. By replacing the product of A with the
one x y := x * y , and applying Proposition 3.3, we are provided with an
absolute-valued real algebra B satisfying x 0 x = llx112e for every x E B
and some fixed idempotent e E B. This implies 112 @ x y @ yyI 2 lly112 for
all x , y E B. Since, in view of Urbanik’s construction, the algebra A (and
hence B ) can be chosen infinite-dimensional, we arrive in Gleichgewicht’s
counterexample 49 to Urbanik’s problem lo3 if every absolute-valued real
algebra A containing a nonzero idempotent and satisfying 11x2+y211 2 lly112
for all x , y E A is isomorphic to R. Finite-dimensional counterexamples are
*@, *H, and *O. The converse of Gieichgewicht’s construction is also true.
Indeed, as proved by Urbanik Io4, if B is an absolute-valued real algebra
such that the linear hull of squares is one-dimensional, then there exists an
absolute-valued real algebra A, with an isometric algebra involution * satisfying xx* = x*x for every x E A , such that B consists of the normed space
of A and the product x @ y := x * y . Gleichgewicht’s absolute-valued infinitedimensional algebras were rediscovered by Ingelstam in a more direct way
(see Proposition 4.4of 54).
:= ( ~ ( ” 1 ) ~
Given an algebra A , let us define inductively x ( l ) := x ,
( ( x , n )E A x N), and let us say that A is semi-algebraic if for every
x E A there exists n E N such that the subalgebra of A generated by x ( ~ )
is finite-dimensional. Clearly, the infinite-dimensional absolute-valued real
algebra B in Gleichgewicht’s counterexample is semi-algebraic. This gives
some interest t o El-Mallah result 44 that If A is an absolute-valued sernialgebraic real algebra fu2filling the requirements in Proposition 3.3, then A
is finite-dimensional.
We conclude this subsection with another result of El-Mallah.
+
Theorem 3.4. 43 Let A be an absolute-valued real algebra endowed with
an isometric algebra involution * such that the equality xx* = x*x holds
f o r every x E A . If A satisfies the identity x ( x x 2 ) = ( x ~ )then
~ , A is
133
isomorphic to either R,C., H,or 0.
Proof. If * is different from the identity operator, then the original proof
in 43 works without problems. Otherwise, A is commutative, and hence
equal to either R, C , or @. (by the commutative Urbanik-Wright theorem).
But does not satisfy the identity z(zz2) = (z2)>".
0
6
3.5. One-sided division algebras
An algebra A is said to be a left- (respectively, right-) division algebra
if, for every nonzero element z E A , the operator L, (respectively, R,)
is bijective. Since absolute-valued one-sided division complex algebras are
equal to C (see Subsection 2.8), our interest centers in the real case. Then,
refining an argument of Wright log,we can prove the following.
Lemma 3.2. Let A be an absolute-valued left-division real algebra. Then
A is a pre-Hilbert space.
Proof. First assume that A has a left unit e. Then, since Le = IA (the
identity operator on A ) , for every norm-one element z E A , we have
Now remove the assumption that A has a left unit, and note that, for each
norm-one element e E A , the normed space of A becomes an absolutevalued algebra with left unit e under the product z @ y := L;'(zy).
It
follows 4 5 llz ell2 11% - ell2 for all norm-one elements e , z E A . Finally,
apply Schoenberg's theorem.
0
+
+
Now we can prove one of the main results in this subsection.
Theorem 3.5. Let A be an absolute-valued real algebra with a left unit e.
Then the absolute-value of A derives from an inner product (.I.), and,
putting x* := 2(zle)e - z, we have (zylz) = (ylz*z) and z*(zy) = J I ~ 1 1 ~ y
for all x , y, z E A .
Proof. We may assume that A is complete. Then an argument, involving
connectedness and elementary Operator Theory, shows that A is a left
134
division algebra (see Lemma 2.2 of 59). By Lemma 3.2, the norm of A
comes from an inner product (. I . ) . For y , u in A with (elu) = 0, we have
(1+ l142)11Y112 = Ile + ~11211Y112= Il(e + U)YIl2
= IIY
+ UY1I2 = (1+ 1 1 ~ 1 1 2 ) 1 1 ~ 1 1 2 + 2(UYlY)
7
and hence (uyly) = 0. By linearizing in the variable y, we deduce
(uy/Iz) = - ( y l u z ) for all u , y , z E A with (el.)
= 0, or, equivalently,
(zylz) = (ylz*z) for all z, y, z E A. Finally, apply Lemma 2.3.
0
The following corollary follows straightforwardly from Theorem 3.5
above, Lemma 2.3 just applied, and the fact that every absolute-valued
left-division algebra is isotopic to an absolute-valued algebra with a left
unit (see the proof of Lemma 3.2).
Corollary 3.3. A n absolute-valued algebra is a left-division algebra i f and
only if it is isotopic to an absolute-valued algebra A whose norm derives
from an inner product (.I.) such that, for each x E A, there exists x* E A
satisfying (xylz) = (ylz*z) for all y, z E A.
Theorem 3.5 and Corollary 3.3 were first proved by the author (see 85
and 8 6 , respectively). The proof of Theorem 3.5 in 85 is different from
that we have given here, and can seem more involved, since Theorem 3.5
is derived there from a more general principle (namely, Theorem 1 of 85).
Really, if we take from the proof of Theorem 1 of 85 the minimum necessary
to get Theorem 3.5, then most complications disappear. From Theorem 3.5
we derive that absolute-valued real algebras with a left unit are left-division
algebras. More generally, we have the following.
Corollary 3.4. A n absolute-valued real algebra A is a left-division algebra
i f (and only if) there exists e E A such that e A = A.
We do not know if Corollary 3.4 remains true when the requirement
e A = A is replaced with the one that e A is dense in A.
In view of Lemma 3.2, absolute-valued left-division real algebras are
composition algebras. In 61, Kaplansky proved that composition division
algebras are finite-dimensional, and commented on his attempts to show
that the same is true when “division” is relaxed to “left-division”. We are
going t o realize that such attempts could not be successful, by constructing
absolute-valued infinite-dimensional left-division real algebras. To this end,
is it convenient to reformulate Theorem 3.5 in a more sophisticated way.
135
We recall the facts, already commented in Subsection 2.2, that smoothnormed real algebras are pre-Hilbert spaces, and that their algebraic structure is well-understood. Some precisions, taken from 8 2 , are needed here.
For instance, if A is a smooth-normed real algebra, then the mapping
x
x* := 2(x(1)1 - x becomes an algebra involution on A , which is
uniquely determined by the fact that, for every x E A, both x x* and x*x
lie in R1. Here 1 denotes the unit of A , and (. I . ) stands for the inner product from which the norm of A derives. If the smooth-normed real algebra
A is commutative, then actually the unit and the inner product determine
the algebra product by means of the equality
.--)
+
XY = (4l)Y + (YI1)X - (XlY)l.
(1)
Now note that, conversely, every nonzero real pre-Hilbert space H becomes
a smooth-normed commutative real algebra by choosing any norm-one element 1E H and then by defining the product according to the equality (1).
Note also that the choice of the norm-one element 1 above is structurally
irrelevant because pre-Hilbert spaces are transitive normed spaces. It follows that smooth-normed commutative real algebras and nonzero real preHilbert spaces are in a bijective categorical correspondence. Now, Let A be
a smooth-normed commutative real algebra, and let H be a nonzero real
pre-Hilbert space. By a unital *-representationof A on H we mean any
linear mapping 4 : A .+ L ( H ) satisfying 4(1) = I H , +(x2) = (4(x))’, and
(+(x)(r])[[)
= (r]Iq5(x*)(<)) for every x E A and all r], E H . The first assertion in Theorem 3.6 immediately below is easily verified (see t~~for details),
whereas the second one is the desired reformulation of Theorem 3.5.
<
Theorem 3.6. I f A is a smooth-normed commutative real algebra, and if
is a unital *-representation of A o n its own pre-Hilbert space, then the
normed space of A with the new product 0 defined by x 0 y := 4(x)(y)
becomes an absolute-valued real algebra with a left unit. Moreover, there
are no absolute-valued real algebras with a left unit others than those given
by the above construction.
One of the main results in the mathematical modelling of Quantum
Mechanics is the possibility of representing the so-called “Canonical Anticommutation Relations” by means of bounded linear operators on complex
Hilbert spaces 14. Applying such a result, it is proved in 85 that every complete smooth-normed infinite-dimensional commutative real algebra has a
unital *-representation on its own Hilbert space. Therefore we have the
following.
136
Theorem 3.7. Every infinite-dimensional real Hilbert space becomes an
absolute-valued algebra with a left unit, under a suitable product.
In the case that the infinite-dimensional real Hilbert space is separable,
Theorem 3.7 was proved simultaneously and independently by Cuenca 2 3 .
Cuenca’s proof isof course easier than the one in 85 for the general case.
The key idea in 23 consists of a “doubling process” which, after an induction
argument, assures that, for every n E N,the smooth normed commutative
real algebra A, of dimension n has a unital *-representation 4, on the real
pre-Hilbert space H , of dimension 2”-l. Moreover, regarding
A1 C_
A2
C ... C A, E ...
and H1
C Hz
... & H , & ...
in a convenient way, we have +,+l(z)(q)= +,(z)(q) whenever n,z,and
q are in W, A,, and H,, respectively. Then A := U n E ~ A nis a smooth
normed commutative real algebra having a unital *-representation on the
real pre-Hilbert space H := U n E ~ H n Since
.
H can be identified with the
pre-Hilbert space of A , the separable version of Theorem 3.7 follows from
the first assertion in Theorem 3.6 by passing t o completion.
The proof of Theorem 3.7 given in 85 shows in addition that the product, converting the arbitrary infinite-dimensional real Hilbert space into an
absolute-valued algebra with a left unit, can be chosen in such a way that
the corresponding algebra has no nonzero proper closed left ideals.
Recently] Elduque and Perez 33 have proved that every infinitedimensional real vector space can be endowed with a pre-Hilbertian norm
and a product which convert it into an absolute-valued algebra with a left
unit. Since, in the construction of 33, the pre-Hilbertian norm and the
product can be chosen in such a way that an arbitrarily prefixed algebraic
basis becomes ortonormal, it follows that Theorem 3.7 can be derived from
the Elduque-PBrez result by an easy argument of completion.
Very recently] relevant progresses about the representations of the
Canonical Anticommutation Relations on separable real Hilbert spaces have
been done in the paper of E. Galina, A. Kaplan, and L. Saal 50. As pointed
out by these authors, their results give rise to a classification, up to an isotopy, of all separable complete absolute-valued left-division real algebras.
Now that the existence of absolute-valued infinite-dimensional leftdivision real algebras is not in doubt, Propositions 3.4 and 3.5, and Corollary 3.5 below have their own interest.
Proposition 3.4. Let A be an absolute-valued real algebra with a left unit.
Then the following assertions are equivalent:
137
(1) For all x,y E A, there exists z E A such that L, o L, = L, o L,
(2) The dimension of A is equal to either 1, 2, or 4.
Proof. Keeping in mind that R, C, and W are associative division algebras,
the implication ( 2 ) + (1) is an easy consequence of Proposition 1.8. Let L
denote the space of all left multiplication operators on A. It follows easily
from Theorem 3.5 that F 2 lies in L whenever F is in L. Therefore both
1
1
F 0G := - ( F o G G o F ) = - ( ( F 3- G)2 - ( F - G)2)
2
8
and
+
F o G OF = 2 F 0 ( F a G ) - F 2 O G
lie in L whenever F , G are in L. Assume that (1) is true. Let z,y be in A
with x # 0. Then, keeping in mind that the operator L, is bijective (by
Theorem 3.5), the assumption (1) reads as L, o L, o L i l E L. But, again
by Theorem 3.5, the norm of A derives from an inner product (.I.) such
that, denoting by e the left unit of A , and putting z* := 2(zle)e- z, we
have Lzl = ~ \ X ) ) - ~ L
. Thus
,.
L, o L, o L,. E L. Since L, o L, o L, E L and
z + x* = 2(xle)e,we deduce (xle)L, o L, E L or, equivalently, L, 0 L, E L
whenever (xle) # 0. Since the set {t E A : (tle) # 0 ) is dense in A, and
the mapping t -+ Lt from A to the normed algebra C ( A ) (of all continuous
linear operators on A ) is a linear isometry, we obtain L, o L, E L without
any restriction. In this way, L becomes a subalgebra of C ( A )containing the
unit of C(A).Since the algebra C ( A ) is associative, and L is a pre-Hilbert
space for the operator norm, it follows from Theorem 3.1 of 54 that L is a
copy of R, @, or W.Therefore A has dimension equal to 1, 2, or 4.
0
Proposition 3.5. Let A be an absolute-valued real algebra with a left unit,
and let 11 . 1 1 be an algebra norm on A . Then we have 11 . I( I (11 . 111.
Proof. Let e denote the left unit of A, and let 2 be in A. According to
Theorem 3.5, we have L,. o L, = 11x1121~,where x* := 2(zle)e - x. It
follows
1I4l2I
lIILx*IIIIIILXIII 5 lll~*llIlIl~Ill
I (2ll~llllIellI+ 111~111>111~111
7
so (1+ lle1112)11x112 5 (111~111 + ll~llllle111)2,
and so
Now apply Proposition 1.1.
( d m - Illelll)ll~llI 111~111*
0
We do not know if Proposition 3.5 remains true whenever the requirement of the existence of a left unit in A is relaxed to the one that A is a
left-division algebra. In any case, we have the following.
138
Corollary 3.5. Let A be a left-division real algebra. Then there exists at
most one absolute value o n A.
Proof. Let 11 . 11 and 111.111 be absolute values on A. Fix e E A with llell = 1,
and consider the absolute-valued real algebra B consisting of the vector
space of A, the norm 11 11, and the product z o y := L;'(zy). Since B
has a left unit, and lllelll-llll . 1 1 is an algebra norm on B , Proposition 3.5
applies giving that )I .I) 5 l)elll-ll]l.)I). Then, keeping in mind that 1. )I/ is an
algebra norm on A, and that 11 11 is an absolute value on A , we deduce from
Proposition 1.1that 11 )I 5 1)) . 111. By symmetry, we have also 1) . 1 ) 5 1) . 11.0
1
Corollary 3.5 above was first proved in
with crafter techniques.
3.6. Automatic continuity
Minor changes to the proof of Corollary 3.5 could allow us to realize that
if A is an absolute-valued left-division algebra, and i f 1 1 . 1 1 is a complete
algebra norm on A , then we have )I . 11 5 1 1 ' 111. However, this fact becomes
unsubstantial in view of the result which follows.
Theorem 3.8. Let A be a complete normed algebm over K, let B be an
absolute-valued algebra over K,and let ; A + B be an algebra homomorphism. Then 4 is contractive.
$J
Keeping in mind Proposition 1.1, the actual message of Theorem 3.8
above is that algebra homomorphisms from complete normed algebras to
absolute-valued algebras are automatically continuous. We do not enter
here the original proof of Theorem 3.8 in 85. Limiting ourselves to mention
its main ingredients (namely, Theorem 2.2 and a little Operator Theory,
including Lemma 3.1 of 83), we prefer to review here how such a proof
has inspired further developments of the theory of automatic continuity in
settings close to that of absolute-valued algebras. To this end, we note that,
replacing the absolute-valued algebra B with the completion of the range
of 4, the proof of Theorem 3.8 reduces to the case that B is complete and
4 has dense range. Thus, for K = @, Theorem 3.8 follows straightforwardly
from Theorem 3.9 immediately below.
Theorem 3.9. Algebra homomorphisms from complete normed complex
algebras to complete normed complex algebras with no nonzero two-sided
topological divisor of zero are continuous.
139
To prove Theorem 3.9, we introduced in
quasi-division algebras.
These are defined as those algebras A such that L, or R, is bijective for
every x E A \ (0). Then we proved that, if A and B are complete normed
algebras over R, if B is not a quasi-division (respectively, division) algebra,
and i f B has no nonzero two-sided (respectively, one-sided) topological divisors of zero, then dense range algebra homomorphisms from A to B are
continuous. With the help of 83, this implies that, i f A and B are complete normed algebras over K,and i f B has no nonzero two-sided topological
divisors of zero, then surjective algebra homomorphisms from A to B are
continuous. Now, Theorem 3.9 follows from the results just quoted and
Proposition 3.6 immediately below.
Proposition 3.6.
Every complete normed quasi-division complex algebra has dimension 5 2.
For JK = R,Theorem 3.8 can be also derived from the results in quoted
above, by applying Wright’s Theorem 1.3 instead of Proposition 3.6. Some
additional information, related to the discussion of the proof of Theorem
3.8 just done, is collected in the next remark.
Remark 3.1. i) In view of Corollary 3.4, absolute-valued quasi-division
algebras are in fact one-sided division algebras.
ii) We do not know if Theorem 3.9 remains true when real algebras
replace complex ones. Even if the range algebra has no nonzero one-sided
topological divisors of zero, the question remains open. The point is that
the old problem log,if every complete normed division real algebra is finitedimensional, remains still unsolved. In relation to this problem, let us note
that, as a consequence of Theorems 3.5 and 3.7, there exist complete normed
infinite-dimensional real algebras A such that, f o r every x E A \ { 0 } , the
operators L, and R, are surjective (see Proposition 8 of 85 for details).
iii) The question of the automatic continuity of homomorphisms into
finite-dimensional algebras has been definitively settled in 17. Indeed, given
a norrned finite-dimensional algebra B over K,all algebra homomorphisms
from all complete normed algebras over K to B are continuous if and only
i f B has no nonzero element x with x 2 = 0.
From now on, let R be a locally compact Hausdorfl topological space.
Given a normed algebra A over JK, the space Co(0,A), of all A-valued continuous functions on Q which vanish at infinity, becomes a normed algebra
over JK under the product defined point-wise and the sup norm. If follows
140
from Lemma 2.10 of 71 and Theorem 3.8 that, if A is a n absolute-valued algebra over K, then algebra homomorphisms from complete normed algebras
over R to Co(R,A) are continuous. The paper of M. M. Neumann, M. V.
Velasco and the author 71, a minor result of which has been just applied,
contains a deeper variant of the fact reviewed above. By an 3-algebra we
mean a real or complex algebra endowed with a complete metrizable vector
space topology making the product continuous. Now we have the following.
Theorem 3.10. 71 Let F be an 7-algebra over R, let A be an absolutevalued algebra over K, and let 4 : F + Co(R,A) be a n algebra homomorphism. Assume that R has no isolated points, and that the range of 4
separates the points of R. Then 4 is continuous.
Here, that a subset S of Co(R,A) separates the points of R means
that, whenever w1 and w2 are different points of 0 , we can find f E S such
that f(wl) = 0 and f(w2)# 0. Other results of a flavour similar to that of
Theorem 3.10 are also proved in 71. For example, if A is a n absolute-valued
algebra over R, if R has n o isolated points, and if F is a subalgebra of
Co(S2,A ) which separates the points of R and is endowed with a n 7-algebra
topology, then every derivation of F is continuous. As a consequence, if
A is a complete absolute-valued algebra over R, and if R has no isolated
points, then every derivation of Co(R, A ) is continuous. We recall that a
derivation of an algebra A is a linear operator D on A satisfying
D(ZY) = a Y ) + D b ) Y
for all IC, y E A. The results in 71 we have reviewed are in fact corollaries to more general facts. In particular, all these results remain true
when absolute-valued algebras are replaced with normed algebras without
nonzero left topological divisors of zero. Appropriate variants of such results
also hold when absolute-valued algebras are replaced with normed algebras
with a unit. The associative side of 71 has its own interest, and appears as
Section 5.6 of 64. The announcement of 71 done in lo7 centres more in the
nonassociative aspects, paying special attention to the applications in the
theory of absolute-valued algebras.
In contrast with Theorem 3.8, we do not know if derivations of complete
absolute-valued algebras are automatically continuous. Anyway, we have
the following.
Proposition 3.7. Absolute-valued complex algebras have no nonzero continuous derivations.
141
Proof. Let A be an absolute-valued complex algebra (which can be assumed complete), and let D be a continuous derivation of A. Then, for
every complex number A, exp(AD) is an algebra automorphism of A , and
hence we have 11 exp(AD)II = 1 (by Proposition 1.1). Now apply Liouville’s
theorem to deduce D = 0.
0
Proposition 3.7 does not remain true when real algebras replace complex ones. Indeed, W and 0 have nonzero derivations in abundance. With
the language of “numerical ranges” 12, the general version of Proposition
3.7 is that continuous derivations of an absolute-waled algebra over K have
numerical ranges equal to zero. Proposition 3.7 then follows since, by
the Bohnenblust-Karlin theorem, continuous linear operators on a complex
normed space must be zero provided their numerical ranges are zero.
4. Some deviations of the theory
4.1. Nearly absolute-valued algebras
For every normed algebra A over K,let us define p ( A ) as the largest nonnegative real number p satisfying pllz\IIIylI 5 llzyll for all z,y E A. Those
normed algebras A such that p ( A ) > 0 are called nearly absolute-valued
algebras. Let A be a normed finite-dimensional algebra over K. Then, by
the compactness of spheres, A is nearly absolute-valued if (and only i f ) it
is a division algebra. Moreover, if this is the case, then A is isomorphic to
C when K = C, and the dimension of A is equal to 1 , 2 , 4 , or 8 when K = R
(by Theorem 1.2). On the other hand, by Hopf’s theorem (see page 235
of 31), nearly absolute-valued finite-dimensional commutative real algebras
have dimension 5 2. We note that, since every finite-dimensional algebra over K can be endowed with an algebra norm, nearly absolute-valued
finite-dimensional real algebras and finite-dimensional division real algebras
essentially coincide.
By Theorem 1.1, every nearly absolute-valued alternative real algebra
is isomorphic to either R, @, W, or 0,so that no much more can be
said about such an algebra. The consequence that nearly absolute-valued
alternative algebras are algebra-isomorphic to absolute-valued algebras is
no longer true if alternativeness is removed (even for finite-dimensional
normed algebras A wit p ( A ) near one). For instance, for 0 < E <
consider the normed real algebra A consisting of the normed space of W
and the product IC 0y := (1 - E)ZY + E ~ Z .Then we have p ( A ) 2 1 - 2 ~ ,
but A cannot be algebra-isomorphic to an absolute-valued algebra. Indeed,
4,
142
A is not associative and has a unit, whereas every absolute-valued fourdimensional real algebra with a unit is isomorphic t o W (by Theorem 2.1).
The above example shows in addition how a theory of nearly absolutevalued algebras parallel to that of absolute-valued algebras cannot be expected. Another notice in the same line is that, in contrast with Theorem 2.3, there exist nearly absolute-valued infinite-dimensional commutative algebras over K. Indeed, as proved in Example 1.1 of 59, for every
infinite set U and every injective mapping 6 : U x U + U ,the complete
normed real algebra A obtained by replacing the product of &(U, 6, K) (see
Subsection 3.1) with x 0y := ; ( x y y z ) satisfies p ( A ) 2 2-a.
Despite the above limitations, in the paper of Kaidi, Ramirez, and the
author 59 just quoted we wondered whether there could be a theory of
nearly absolute-valued algebras “nearly” parallel to that of absolute-valued
algebras. More precisely, we raised the following.
+
Question 4.1. Let P be anyone of the purely algebraic properties leading
absolute-valued real algebras to the finite dimension. Is there a universal
constant 0 5 K p < 1 such that every normed real algebra A satisfying P
and p ( A ) > K p is finite-dimensional?
We were able to answer Question 4.1 for the most relevant choices of
Property P. Thus we have the following.
Theorem 4.1. 59 Question 4.1 has an afirmative answer whenever Property P is equal to the existence of a unit, the commutativity, or the algebraicity. Moreover, for such choices of P , the universal constant K p
can be (uniquely) chosen in such a way that there exists a normed infinitedimensional real algebra satisfying P and p ( A ) = K p .
Now, fundamental Theorems 2.1, 2.3, and 2.11 are “nearly” true when
nearly absolute-valued algebras replace absolute-valued algebras. As a consequence, a normed power-commutative real algebra A is finite-dimensional
whenever p ( A ) is near one. Many other results of the same flavour can
be obtained (see for example the variants of Theorems 2.2, 2.8, and 3.8
proved in Corollaries 3.2 and 3.4, and Theorem 3.3 of 59, respectively). We
already know that K p 2 2 - i when P means commutativity. When P
means algebraicity or existence of a unit, we do not know whether or not
the equality K p = 0 holds. In any case, Nearly absolute-valued complex
algebras are isomorphic to C whenever they have a left unit or are algebraic
(see Remark 2.8 of 59 and our Subsection 2.8, respectively). If they are
commutative, then a result similar to the one given by Theorem 4.1 (with
143
P equal to the commutativity) holds.
A normed space X is said to be uniformly non-square if there exists
0 < u < 1 such that the inequality min{l)z y J JJ]z
, - yII} < 2 0 holds
for all x,y in the closed unit ball of X . We note that pre-Hilbert spaces
are uniformly non-square, and that the completion of every uniformly nonsquare normed space is a superreflexive Banach space (see Theorem VII.4.4
of 27). We also remark that neither absolute-valued algebras nor nearly
absolute-valued finite-dimensional algebras with a unit need be uniformly
non-square (by our Subsection 3.1 and Example 2.1 of 59, respectively).
These facts give its own interest to the following variant of Theorem 3.5.
+
Theorem 4.2. 59 Let A be a normed real algebra with p ( A ) > 2-a. If there
exists a E A such that a A is dense in A , then A is uniformly non-square.
4.2. Other deviations
By a trigonometric algebra we mean a pre-Hilbert real algebra A satisfying llzy1I2 + (4Y)2 = 11~11211Y112(or, equivalently, IlzYIl = ll~llllYllsin%
where Q is the angle between z and y ) for all z, y E A \ (0). By cleverly applying Hurwiz’s theorem (see Subsection 2.2), P. A. Terekhin looshows that
the dimensions of finite-dimensional trigonometric algebras are precisely 1,
2, 3, 4, 7, and 8. The existence of complete trigonometric algebras of arbitrary infinite Hilbertian dimension is implicitly known in 59. Indeed, if U is
an infinite set, and if 6 is an injective mapping from U x U to U , then the
real Hilbert algebra obtained from the absolute-valued algebra C2(U,6, IR)
(see Subsection 3.1) by replacing its product with the one x @ y := Iy--y2
4
becomes a trigonometric algebra (see Remark 1.6 of 59 for details). Actually, all infinite-dimensional -trigonometric algebras can be constructed in
a transparent way from the absolute-valued real algebras with involution
considered in Subsection 3.4 (see 9).
By a triple system over a field IF we mean a nonzero vector space
T over IF endowed with a trilinear mapping (. . .) : T x T x T 4 T .
Absolute-valued triple systems over K are defined as those triple systems T over K endowed with a norm )I . 11 satisfying [I(zyz)ll= llzllIlyllllzll
for all z, y , z E T . Each absolute-valued triple system gives rise to “many”
absolute-valued algebras. Indeed, if T is an absolute-valued triple system,
and if u is a norm-one element in T, then the normed space of T becomes
an absolute-valued algebra under the product x 0y := ( m y ) . As pointed
out in 18, this implies (by Proposition 1.5) that the n o r m of every absolutevalued finite-dimensional real triple system springs f r o m a n inner product.
144
It follows that, if T is an absolute-valued finite-dimensional real triple system, then the mapping q : x --f ) ) x ) )is2 a quadratic form on T satisfying
q ( ( x y z ) ) = q ( x ) q ( y ) q ( z )for all x,y , z E T . Now we can mimic Albert’s
definition of isotopy (see Subsection 1.3), and apply the main result in
K. McCrimmon’s paper 67, to get that, u p to a n isotopy, the absolutevalued finite-dimensional real triple systems are R,C (with ( x y z ) = x y z in
both cases), W (with (xcyz) equal to either x y z , x z y , or y x z ) , and 0 (with
( x y z ) equal to either ( x y ) z , ( x z ) y , ( y x ) z , ~ ( g z )~, ( z y ) or
, y ( x z ) ) . This
result is a sample of how the study of absolute-valued triple systems can
be promising (see l8 and ‘O).
An algebra A is said to be two-graded if it can be written as
A = A0 @ A l l where A0 and A1 are nonzero subspaces of A satisfying
AiAj Ai+j for all i , j E Zz. Two-graded absolute-valued algebras
are defined as those normed two-graded algebras A = A0 @ A1 over K satisfying IlxixjII = Ilxillllxjll for all i , j E Zz and every (xi,xj)E Ai x Aj.
The work of A. J. Calder6n and C. Martin l9 deals with these objects, and
starts with the observation that, if A is a two-graded absolute-valued algebra, then, in a natural way, A0 i s a n absolute-valued algebra, and A1 i s a n
absolute-valued triple system. As a consequence, two-graded absolute-valued
finite-dimensional real algebras have dimension 2, 4, 8, or 16.
5. Absolute-valuable Banach spaces
In this concluding section we are going to deal with those Banach spaces
which underlie complete absolute-valued algebras. Such Banach spaces will
be called absolute-valuable. The finite-dimensional side of this topic
is definitively solved by Albert’s Proposition 1.5. Indeed, the absolutevaluable finite-dimensional real Banach spaces are precisely the real Hilbert
spaces of dimension 1, 2, 4, and 8. On the other hand, it is clear that
C is the unique absolute-valuable finite-dimensional complex Banach space.
Therefore, the interest of absolute-valuable Banach spaces centers into the
infinite-dimensional case.
5.1. The isometric point of view
We already know that, for every infinite set U , the classical Banach spaces
lp(U,K) (1 5 p < co) and Q(V,JK) are absolute-valuable (see Subsection 3.1). In fact, the rol played there by JK can be also played by any
absolute-valued algebra, and hence we have that, given a n infinite set U
and a Banach space X , the Banach spaces l p ( U , X ) (1 5 p < m) and
145
co(U,X ) are absolute-valuable whenever so is X . Even, the value p = 00 is
allowed above (by Proposition 5.2 below). Other stability properties of the
class of absolute-valuable Banach spaces can be also derived from previously
reviewed results. For example, it follows from Lemma 2.4 that the normed
ultraproduct of every ultrafiltered family of absolute-valuable Banach spaces
is a n absolute-valuable Banach space. More examples of absolute-valuable
Banach spaces are given in Proposition 5.1 immediately below. As usual,
given Banach spaces X and Y over K,we denote by C ( X ,Y ) the Banach
space over R of all bounded linear operators from X to Y, and by K ( X , Y)
the closed subspace of C ( X lY) consisting of all compact operators from
X to Y. Moreover, we write X * , L ( X ) , and K ( X ) instead of L ( X , R ) ,
C ( X ,X ) , and K ( X , X ) , respectively.
Proposition 5.1. Let 1 5 p 5 m, let U1 be a n infinite set, and let X 1
stand f o r lP(U~,IK).
Then X,* is absolute-valuable. Moreover, if U2 is
another infinite set, and if X2 stands f o r e P ( U 2 , R ) , then C ( X l , X , ) and
K ( X 1 ,X 2 ) are absolute-valuable.
As a consequence, infinite-dimensional Hilbert spaces over R are
absolute-valuable, and moreover, i f H1 and H2 are infinite-dimensional
Hilbert spaces over IK, then C ( H 1 ,H z ) and K ( H 1 , H2) are absolute-valuable.
The paper of J. Becerra, A. Moreno, and the author 7 , from which we
have taken Proposition 5.1, also contains Theorem 5.1 immediately below.
Given a topological space El we denote by dens(E) the density character
of E (see Subsection 3.2).
Theorem 5.1. Every Banach space X over R is linearly isometric to a
subspace of a Banach space Y over R with dens(Y) = dens(X) and such
that Y, Y*, C(Y), and K(Y) are absolute-valuable.
The ideas in the proof of Theorem 5.1 are not far from those in Proposition 5.2 immediately below. I n what follows, R will denote a compact
Hausdorff topological space.
Proposition 5.2. Assume that there exists a continuous surjection from
R to R x R, and let X be an absolute-valuable Banach space. Then C(R, X )
is absolute-valuable.
Proof. Let us choose a product (x,y) 4 zy on X converting X into an
absolute-valued algebra, let q5 : R + R x R stand for the continuous surjection whose existence is assumed, and, for i = 1,2, let 7ri : R x R 4 R
146
denote the i-th coordinate projection. Then the product o on C(R,X)
defined by (f o g ) ( w ) := f(7rI(q5(w)))g(7rz($(w)))(for every w E R and all
f,g E C(R, X ) ) converts C(R, X)into an absolute-valued algebra.
We note that the choice R = [0,1] is allowed in Proposition 5.2 (see
and references therein). According to that proposition, the existence of a
continuous surjection from R to R x R is a suficient condition for C(R, K)
to be absolute-valuable. In 69, a partial converse is shown. Indeed, we have
the following
Theorem 5.2. If C(fl,K) is absolute-valuable, then there exists a closed
subset F of R, and a continuous surjection from F to R x 52.
As a consequence, C(R,K) is not absolute-valuable whenever R is the
one-point compactification of a discrete infinite space (a fact first proved
in 7). In particular, the classical space c of all real or complex convergent
sequences is not absolute-valuable. Theorem 5.2 is also applied in 69 to
prove that, in the case that R is metrizable, C(R,K) is absolute-valuable i f
and only if R is uncountable. The arguments for Theorem 5.2 mimic those
in the proof of a theorem of W. Holsztynski on nonsurjective isometries
between C(R)-spaces (see Section 22 of 96).
Given a Banach space X , we denote by Q the group of all surjective
linear isometries on X . We recall that a Banach space X is said to be transitive (respectively, almost transitive) if, for every (equivalently, some)
norm-one element u in X , B(u) is equal to (respectively, dense in) the unit
sphere of X . The reader is referred to the book of S. Rolewicz 92 and the
survey papers of F. Cabello l5 and Becerra-Rodriguez * for a comprehensive
view of known results and fundamental questions in relation to the notions
just introduced. Hilbert spaces become the natural motivating examples of
transitive Banach spaces, but there are also examples of non-Hilbert almost
transitive separable Banach spaces, as well as of non-Hilbert transitive nonseparable Banach spaces. However, the Banach-Mazur rotation problem,
if every transitive separable Banach space is a Hilbert space, remains unsolved to date. Since almost transitive finite-dimensional Banach spaces
are indeed Hilbert spaces, the rotation problem is actually interesting only
in the infinite-dimensional setting. Then, since infinite-dimensional Hilbert
spaces are absolute-valuable, we feel authorized to raise the following strong
form of the Banach-Mazur rotation problem.
Problem 5.1. Let X be an absolute-valuable transitive separable Banach
space. Is X a Hilbert space?
147
We hope Problem 5.1 to have an affirmative answer in the next future.
In the meantime, we must limit ourselves to review the following.
Proposition 5.3.
There exists a non-Hilbert absolute-valuable almost
transitive separable Banach space X such that C(X, Y) and K(X,Y) are
absolute-valuable for every absolute-valuable Banach space Y .
Actually, the space X in Proposition 5.3 can be taken equal to
L1([0,1])). Proposition 5.3 implies (applying Lemma 2.4 among other tools)
that there exists a non-Hilbert absolute-valuable transitive non-separable
Banach space. One of the tools in the proof of Proposition 5.3 is the following.
Lemma 5.1. Let X and Y be Banach spaces over K. Assume that the
complete projective tensor product XGTX is linearly isometric to a quoY)
tient of X, and that Y is absolute-valuable. Then C(X, Y) and F(X,
are absolute-valuable. Here 3 ( X ,Y) stands f o r the space of all finite-rank
operators f r o m X to Y.
We conclude the present subsection by applying Lemma 5.1 to prove
the following
Theorem 5.3. Every Banach space X over JK is linearly isometric to
a quotient of an absolute-valuable Banach space Y over JK satisfying
dens(Y) = dens(X), and such that L(Y,Z), and Ic(Y,Z) are absolutevaluable for every absolute-valuable Banach space Z over K.
Proof, Let U be a set whose cardinal number equals dens(X), and let Y
stand for the absolute-valuable Banach space l l ( U ,K). Clearly, we have
dens(Y) = dens(X). On the other hand, it is well-known that X is linearly
isometric to a quotient of Y. (In fact, noticing that X becomes a complete
normed algebra under the zero product, we had to show a little more when
we proved Corollary 3.2.) Finally, noticing that
YG,,Y = lI(U,JK)&ll(U,JK) = l l ( U x U,K) = l,(U,K) = Y
(see Ex 3.27 of ") and that Y* has the approximation property (see 5.2
of 28, the proof is concluded by applying Lemma 5.1 (see 5.3 of 28).
0
5 . 2 . The isomorphic point of view
Most isomorphic properties on Banach spaces considered in the literature
are inherited by quotients and/or subspaces. Therefore, by Theorem 5.1
148
and Corollary 5.3, none of such properties can be implied by the absolute valuableness. Now, recall that a Banach space X is called weakly
E~
countably determined if there exists a countable collection { K n } nof
w*-compact subsets of X** in such a way that, for every x in X and every
u in X**\ X, there exists no such that x E K,, and u @ Kn0. If X is
either reflexive, separable, or of the form c o ( r ) for any set r, then X is
weakly countably determined. In fact, the class of weakly countably determined Banach spaces is hereditary, and contains the non hereditary class
of weakly compactly generated Banach spaces (see Example VI.2.2 of 29 for
details). Among the results proved in concerning the isomorphic aspects
of absolute-valuable Banach spaces, the main one is the following.
Theorem 5.4. Every weakly countably determined real Banach space, different from R, is isomorphic to a real Banach space X such that both X
and X* are not absolute-valuable.
We do not know if the requirement of countable determination can be
removed in Theorem 5.4.
A Banach space X is said to be hereditarily indecomposable if, for
every closed subspace Y of X , the unique complemented subspaces of Y
are the finite-dimensional ones and the closed finite-codimensional ones.
According to the paper of W. T. Gowers and B. Maurey 51, the existence
of infinite-dimensional hereditarily indecomposable separable reflexive Banach spaces over K as not in doubt. On the other hand, we have proved in
that infinite-dimensional hereditarily indecomposable Banach spaces over R
fail to be absolute-valuable. Thus, since the hereditary indecomposability is
preserved under isomorphisms, we are provided with an infinite-dimensional
Banach space over R which is not isomorphic to any absolute-valuable Banach space. In other words, the property of absolute valuableness is not
isomorphically innocuous. In the case K = C, more can be said. Indeed,
we have the following.
Proposition 5.4. Let X be an infinite-dimensional hereditarily indecomposable complex Banach space. Then X cannot underlie any complete
normed algebra without nonzero two-sided topological divisors of zero.
Proof. By Corollary 19 of 51, X is not isomorphic to any of its proper
subspaces. Assume that, for some product, X is a complete normed algebra
without nonzero two-sided topological divisors of zero. Then, for every
x E X \ {0}, L, or R, is an isomorphism onto its range, and hence it is
149
bijective. Now, X is a quasi-division algebra, and hence finite-dimensional
(by Proposition 3.6).
Let us say that a Banach space X over K satisfies the Shelah-Steprans
property whenever X is not separable and, for every F E L ( X ) ,there exist
X = X(F) E IK and S = S ( F ) E L ( X ) such that S has separable range and
the equality F = S Xlx holds. In our present discussion, Banach spaces
enjoying the Shelah-Steprans property play a rol similar to that of infinitedimensional hereditarily indecomposable Banach spaces. Indeed, reflexive Banach spaces satisfying the Shelah-Steprans property do exist (see 97
and lo8),the Shelah-Steprans property is preserved under isomorphisms,
and Banach spaces over W fulfilling the Shelah-Stepmns property fail to be
absolute-valuable '. By the way, the proof of the result of just reviewed
can be slightly refined to get the following.
+
Proposition 5.5. Let X be a Banach space over K satisfying the ShelahSteprans property. Then X cannot underlie any complete normed algebra
without nonzero two-sided topological divisors of zero.
Proof. First, we note that, for F in L ( X ) , the couple ( X ( F ) , S ( F )given
)
by the Shelah-Steprans property is uniquely determined, and that the mappings X : F -+ X ( F ) and S : F -+ S ( F ) from L ( X ) to W and L ( X ) , respectively, are linear. Now, since X is not separable, and ker(X) consists of those
elements of L ( X ) which have separable range, we have X ( F ) # 0 whenever
the operator F on X is an isomorphism onto its range. Assume that, for
some product, X is a complete normed algebra without nonzero two-sided
topological divisors of zero. Then, since L, or R, is an isomorphism onto
its range whenever 5 is in X \ {0}, it follows that the linear mapping
x -+ (X(L,),X(R,)) from X to K2 is injective. Therefore X is finite0
dimensional, a contradiction.
Concerning the topic of the present section, let us say that the study of
absolute-valuable Banach spaces is just started, so that there are more problems than results on the field. Since we have mainly emphasized the results,
let us conclude the paper with one of the problems non previously collected.
Let us say that a Banach space is nearly absolute-valuable if it underlies some complete nearly absolute-valued algebra (see Subsection 4.1). It
is easy to see that the near absolute valuableness is preserved under isomorphisms. Consequently, isomorphic copies of absolute-valuable Banach
spaces are nearly absolute-valuable. However, we do not know whether
150
or not every nearly absolute-valuable Banach space is isomorphic t o an
absolute-valuable Banach space.
Acknowledgements
The author thanks the organizers of the ‘$First International Course on
Mathematical Analysis in Andalucia”, Professors A. Aizpuru and F. Leon,
for inviting him t o deliver lectures in that Course and write the paper
which concludes now. The discussions began during the Course with Professor J. Benyamini enriched very much the papers and 69. Consequently,
since the review of such papers done in Section 5 above has benefited from
such an enrichment, special thanks are due to him. The author is also
grateful t o J. Becerra, M. Cabrera, J. A. Cuenca, A. Fernfindez, A. Kaidi,
A. Kaplan, J. Martinez, C. Martin, M. Martin, J. F. Mena, A. Moreno,
R. Payfi, A. Peralta, M. I. Ramirez, A. Rochdi, A. Slin’ko, M. V. Velasco,
and the anonymous referee. In one or other way, all they have helped him
very much in the writing of this paper.
’
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