Banach Spaces With Unique Unconditional Basis, Up to Permutation

119 Pages·1985·8.717 MB·English

by  J. Bourgain, P. G. Casazza, J. Lindenstrauss

Memoirs of the American Mathematical Society
Number 322
J. Bourgain, P. G. Casazza,
J. Lindenstrauss and L. Tzafriri
Banach spaces with a
unique unconditional basis,
up to permutation
Published by the
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island, USA
March 1985 • Volume 54 • Number 322 (fourth of 6 numbers)
AMS(MOS) Subject Classification (1970). 46B99, 46E30, 46E99, 47B55.
Library of Congress Cataloging-in-Publication Data
Main entry under title:
Banach spaces with a unique unconditional basis, up to permutation.
(Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 322)
"March 1985, volume 54, number 322 (fourth of 6 numbers)."
Bibliography: p.
1. Banach spaces. 2. Bases (Linear topological spaces) I. Bourgain, Jean, 1954- .
IL American Mathematical Society. III. Series.
QA3.A57 no. 322 [QA322.2] 510 s [515.7'32] 84-28116
ISBN 0-8218-2323-X
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Table of Contents
0. Introduction. 1
1. Unconditional bases of finite direct sums of Banach spaces. 6
2. Infinite direct sums of Hilbert spaces. 12
3. Infinite direct sums of £ f-spaces in the sense of c Q, Part I. 22
4. Infinite direct sums of £ f spaces in the sense of CQ, Part II. 29
5. Infinite direct sums in the sense of £ - 46
2
6. Prime spaces. 49
7. Tsirelson's space. 55
OS OO
8. Complemented subspaces of ( £ ©£j?)i and ( £ ®&2) 75
Y
n=l n=l
9. "Large" subspaces of (& 8d © • • • ffi£ © • • • ) . 84
q q g p
10. Complemented subspaces of (£ © £ 2 © ' ' ' © ^ 2 ©• •) 1 96
2
and (c ©c ©- • ©c © • • )t
0 0 0
11. Open problems. 103
12. REFERENCES 109
iii
ABSTRACT
In this M error we examine the question which Banach spaces have a unique
unconditional basis, up to equivalence and permutation. W e solve this question for
some infinite direct sums of classical sequence spaces and for a Tsirelson type space.
W e also classify, up to isomorphism all the conplemented subspaces of sorre of tte
examples mentioned above.
Key words: unconditional basis, uniqueness up to equivalence and pemutation, direct
sums of classical Banach spaces, Tsirelson space.
0. Introduction.
It is well known that c H\ and & are the only Banach spaces
0t 2
which, up to equivalence, have a unique normalized unconditional
basis (cf. [20], [22],[28] and [25]). This means that if X is one of the
spaces c ,H\ or £ and if \x \~ and {y \£ are two normalized uncon
0 2 n =l n =1
ditional bases of X then the mapping T, defined by Tx = y for all n,
n n
extends to a linear automorphism of X. Any other space with an uncon
ditional basis fails to have this property.
The first (and trivial) step in proving that any space X with a
normalized unconditional basis j^r |^°- which has the uniqueness pro
n =1
perty mentioned above must be either c ,£ or £ , is to observe that the
0 x 2
basis fan 1^=1 has to be equivalent to all its permutations i.e. has to be
symmetric. This remark, as well as the fact that an unconditional basis
by its very definition does not have to come with a prescribed order,
shows that, for unconditional bases which are not symmetric, the
more natural equivalence property is that of equivalence up to permu
tations. It is therefore interesting to discover which spaces X have
a unique normalized unconditional basis, up to equivalence and per
mutation. In other words, for which spaces X it is true that, when
ever \x \~~ and \y \£=\ are two normalized unconditional bases of
n x n
X, there is a permutation n of the integers so that the map T, defined by
Tx = y ( ) for every n, extends to an automorphism of X.
n w n
This question of uniqueness, up to equivalence and permutation,
was first studied by Kothe and Toeplitz in 1934 [20]. The central role
which was attached to this problem by both authors over the years was
recently emphasized in Kothe's talk given during the Toeplitz
1 Received by the editors March 5, 1984.
The second named author was supported in part by NSF MCS-8002221 and MCS-8102238.
The third named author was supported in part by NSF HCS-8102714.
The fourth named author was supported in part by NSF MCS-7903042.
1
2 BOURGAIN. CASAZZA. IJNDENSTRAUSS AND TZAFRIRI
Memorial Conference [19]. This problem is also mentioned by
Mitjagin [31]. As pointed out in [19] and [31] this question was treated
in several papers in the context of Frechet spaces (mainly, nuclear
spaces) and more general locally convex spaces, and the informa
tion available by now is extensive. Of course, since there is no natural
notion of "normalization" in spaces which are not normed, equivalence
up to permutation has to be denned somewhat differently in general
sequence spaces (the bases Jx {~ and \y \™ are said to be
n =1 n =x
equivalent, up to permutation if there is a permutation TX and scalars
^ Jn=i so that the map T, defined by Tx = t y ^ for all n, extends to
n n n n
an automorphism.) In the setting of Banach spaces this question
has received only little attention till now. Besides the joint characteri
zation of c ,£ and £ , mentioned above, the only result on this topic
0 x 2
seems to be that of Edelstein and Wojtaszczyk [8]. They prove that the
spaces c ® £ ,c ® £ ,£ ® £ and c ®£ 0£ have a unique normalized
0 x 0 2 x 2 o 1 2
unconditional basis, up to equivalence and permutation.
In Section 1 of the present paper, we make some general observa
tions on unconditional bases in finite direct sums of Banach spaces.
This enables us in particular to give a new and simple proof of the
result of Edelstein and Wojtaszczyk, mentioned above. We also state and
prove a local version of this result.
The main purpose of the present paper is to consider infinite direct
sums of the spaces c ,£ and £ . It turns out that the treatment of
0 1 2
infinite direct sums is much more complicated. In contrast to the
case of finite direct sums, the infinite direct sums exhibit a quite
surprising behaviour.
In Section 2, we show that the spaces (£ ©£ ©- " ©£2©' ' ' )o
2 2
and (£ ©£ ©- • ©£ ® • • • ) have indeed a unique normalized uncon
2 2 2 1
ditional basis, up to equivalence and permutation.
The case of (£ © £ ffi • • • © £ ® • • • ) is much harder. The proof
t x x 0
that this space (as well as ( C Q S C Q ^ - ^ C Q © - - )^ has, up to
UNIQUE UNCONDITIONAL BASIS 3
equivalence and permutation, a unique normalized unconditional basis
requires, besides the arguments presented in Section 2, several addi
tional results which are presented in Sections 3 and 4. Section 3 is
devoted just to the proof of one auxilliary proposition; the uniqueness
theorem itself is proved in Section 4. As explained at the end of
Section 4, the source of the difficulty of the case
(d ®d® •• ©^i®* •• )> as compared to (£ © £ ® •• • © £ ® '' ' )o>
1 l 0 2 2 2
lies in the following fact. In (£ ©£2 ® ' ' ' © £2 © * ' ' )o. tlle operator 7\
2
which exhibits the equivalence between a K- unconditional basis
and the unit vector basis, can always be chosen so that )| 7"11 || T"1]] is
bounded by a power of K. On the other hand, there are examples of
/^-unconditional bases in (£ © £ i © - • • © £ ® • • ) , for which
x x 0
II T || || T~x1| has to grow exponentially as a function of K, as K -» «>.
Surprisingly, the spaces (£ ® £ © • • • ® £ © • • ) and
x x x 2
(co©co®' ' ©co©" ' * )2 foil t° have a unique normalized uncondi
tional basis, up to equivalence and permutation. This is proved in Sec
tion 5.
In Section 6 we prove a result which is closely related to the
joint characterization of c di and £ , mentioned above. It is shown
0l 2
that these spaces are also the only prime Banach spaces with a unique
normalized unconditional basis, up to equivalence and permutation.
In order to state the result of Section 6 somewhat differently, let
us introduce the following notation. We say that a Banach space X with
an unconditional basis is of genus n if in it and in all its complemented
subspaces with an unconditional basis the normalized unconditional
basis is unique, up to permutation and equivalence, and if there are
n different isomorphism types of complemented subspaces with an
unconditional basis. The main result of Section 6 shows, in fact, that
the spaces CQ^J and £ are the only spaces of genus 1. The results
2
of Sections 2-4 show that ( £ ®&£)o,{ £ ®gDo ar*d their duals are of
n = l n = l
genus 2. These may be the only Banach spaces of genus 2. The results
of Section 6 give some indication in this direction (we discuss this
4 BOURGAIN. CASAZZA. UNDENSTRAUSS AND TZAFRIRI
briefly in Section 11). It is conceivable that the spaces of finite genus
coincide with the class obtainable from Hilbert spaces (of finite or
infinite dimensions) by taking repeated (finite or infinite) direct sums in
the sense of c or H Our knowledge thus far is not sufficient for
0 v
attacking this question in its generality.
In Section 7 we prove that there exist Banach spaces of infinite
genus. We show that the 2-convexification T^ of Tsirelson's space T (cf.
[36], [9], and [14]) is such a space. This result indicates that it is not
reasonable to expect to get a functional representation (or
classification) of all the spaces having a unique unconditional basis, up
to equivalence and permutation. In Section 7 we also present a charac
terization of the complemented sublattices of T and T&\
The results of Sections 2-4 give in particular a classification of
the complemented subspaces of (£ ©&z © ' ' ' © ^2© ' ' ' )o-
2
{ZxQlxQ- ®d ®' • • ) , (c ©c ©- • ©c ©- • )i> and
1 0 0 0 0
(£ ffi£ ©- • ®g ©' ' ' )i having an unconditional basis. In Sections 8-
2 2 2
10 we reprove this classification in a completely different way and
show, moreover, that every complemented subspace of these spaces
has an unconditional basis. The results and methods of Sections
8-10 do not involve the question of uniqueness of unconditional bases
and, thus, they do not reprove the main point in Sections 2-4. The
classification of all the complemented subspaces of
(£ ffi£ ffi- • ®£ © • • )jo> 1 <p < °°, was done previously by Odell [32].
2 2 2
The main novelty of the discussion of Sections 8-10 is concentrated
in Section 8, where the complemented subspaces of ( £) ©^^Oi and
71 = 1
oo
(2 ©£2)i a r e classified (in Odell's case the corresponding question is
n=i
as
trivial since ( 2 ©^^,1 <p < °°, is isomorphic to & , cf. [33]). Sec-
p
71 = 1
tions 9 and 10 are devoted to the modification needed in Odell's argu
ment so as to adapt it to the situation discussed here.
UNIQUE UNCONDITIONAL BASIS 5
Essentially, all of Sections 2-4 as well as 8-10, of this Memoir, are
devoted to a detailed study of one Banach space, namely
(«i©£i©- • ©£i©- • • )o. (the results on (£ ©£ © • • • ©£ ©' ' ' )o can
2 2 2
be also considered as necessary steps in the study of
(£ © d j © • • © Z! © • • • ) ). It turns out that this is quite a hard space
1 0
to analyze. It is also an important space as can be witnessed by the
role of this space and its non separable analogue
(£ i © £ i © • • • © £ i © • • • )„ in the theory of Banach lattices, in gen
eral, and injective lattices, in particular (cf. [5],[l2],[27]). It was the
study of injective lattices in [27] which directly motivated and led to
this work.
Section 11, which concludes this Memoir, contains a relatively long
list of open problems. Some of these problems are of interest also out
side the framework of the present study.
The main results of Sections 1-5 (with an outline of the proofs)
were presented in two Seminar notes [4],[21].