#### Abstract

We show that the -convexified Tsirelson space for and all its complemented subspaces with unconditional basis have unique unconditional basis up to permutation. The techniques involved in the proof are different from the methods that have been used in all the other uniqueness results in the nonlocally convex setting.

#### 1. Introduction: The Problem of Uniqueness of Unconditional Basis

If is a quasi-Banach space (in particular a Banach space) with a normalized unconditional basis (i.e., for all ), is said to have a *unique unconditional basis (up to permutation)* if whenever is another normalized unconditional basis of , then is equivalent to (after a permutation); that is, there exists an automorphism of which takes one basis to (a permutation of) the other.

The problem of uniqueness of unconditional basis is classical. It is well known that , , and have a unique unconditional basis and that any other Banach space with an unconditional basis fails to have this property [1–3].

If an unconditional basis is unique, in particular it must be equivalent to all its permutations and hence must be symmetric. Thus, the obvious modification for spaces whose canonical basis is unconditional but not symmetric is to require uniqueness of unconditional basis via a permutation, which in many ways is a more natural concept for unconditional bases. Classifying those Banach spaces with unique unconditional bases up to permutation, however, has turned out to be a much more difficult task. The first step toward this classification was taken in 1976 by Edelstein and Wojtaszczyk [4], who showed that any finite direct sum of , , and had that property. After their work, Bourgain, Casazza, Lindenstrauss, and Tzafriri embarked on a comprehensive study aimed at classifying those Banach spaces with unique uconditional basis up to permutation that culminated in 1985 with their AMS* Memoir* [5]. They considered infinite direct sums of the Banach spaces with unique unconditional basis and showed that the spaces , , , and all have unique unconditional bases up to permutation, while, surprisingly, and do not.

However, all hopes for a satisfactory classification were shattered when they discovered that a space of a totally different character, a certain variant of Tsirelson space , also had a unique unconditional basis up to permutation. More recently, further examples of “pathological” spaces with unique unconditional basis up to permutation have been given in [6, 7].

In the context of quasi-Banach spaces which are not Banach spaces, the uniqueness of unconditional basis seems to be the norm rather than the exception. For instance, it was shown in [8] that a wide class of nonlocally convex Orlicz sequence spaces, including the spaces for , have a unique unconditional basis. The same is true in nonlocally convex Lorentz sequence spaces [9, 10] and in the Hardy spaces for [11].

Analogously, it seemed only natural to translate the question of uniqueness of unconditional basis up to permutation to the setting of nonlocally convex spaces that are infinite direct sums of the classical quasi-Banach spaces with a unique unconditional basis, namely, , , , , , , and , where . With the exception of that remains elusive, the uniqueness of unconditional basis has been established for all the other spaces (see [10, 12–14], in chronological order). Although the proofs are very different depending on each case, all of them rely *explicitly* on the corresponding results for their respective Banach envelopes shown in [5] and revolve around the all-pervading “large coefficient” technique ([10,Theorem 2.3]) for establishing the equivalence of basic sequences.

In this paper we change the strategy, and the proofs in Section 2 hinge on Theorem 1.1. This is a general result on complemented unconditional basic sequences in lattice anti-Euclidean quasi-Banach spaces which extends a result from [7]. We recall that a quasi-Banach lattice is called *sufficiently lattice Euclidean* if there is a constant so that for any there are operators and so that , , and is a lattice homomorphism. This is equivalent to asking that is finitely representable as a complemented sublattice of . We will say that is * lattice anti-Euclidean* if it is not sufficiently lattice Euclidean. We will also use the term* sequence space* to mean a quasi-Banach space of sequences so that the canonical basis vectors form a -unconditional basis.

Theorem 1.1 (see [14, Theorem 3.4]). *Let and be quasi-Banach sequence spaces. Suppose that is -convex for some and that is isomorphic to a complemented subspace of . Suppose that the Banach envelope of is lattice anti-Euclidean. Then there exists and a complemented disjoint positive sequence in that is equivalent to the unit vector basis in . Furthermore, the projection of onto may be given in the form
**
where and for all .*

To help determine whether two unconditional bases are permutatively equivalent we will use the following form of the Cantor-Bernstein principle.

Theorem 1.2 (see [11, Proposition 2.11]). *Suppose and are two unconditional basic sequences of a quasi-Banach space then and are equivalent (up to permutation) if and only if is equivalent (up to permutation) to a subsequence of and is equivalent (up to permutation) to a subsequence of .*

#### 2. Main Result

In this section we settle the question of uniqueness of unconditional basis in the spaces for . These spaces are the nonlocally convex counterpart to the -convexification of Tsirelson space for , introduced by Figiel and Johnson in [15]. Those readers who are unfamiliar with Tsirelson space will find a handy construction and some of its elementary properties in [16]. For a more in-depth approach, the standard reference is [17] (cf. [15]).

Given , the -convexification of Tsirelson space is obtained from by putting

for those sequences of real numbers such that . Equation (2.1) defines a norm for and a -norm when . Obviously, the space is simply .

The canonical unit vectors form a -unconditional basis of for all . Casazza and Kalton established in [7] the uniqueness of unconditional basis up to permutation of and its complemented subspaces with unconditional basis as a byproduct of their study of complemented basic sequences in lattice anti-Euclidean Banach spaces. Their result answered a question by Bourgain et al. in [5], where the authors had proved the uniqueness of unconditional basis up to permutation in the -convexification of .

Here we show that for , the space and its complemented subspaces with unconditional basis belong to the class of quasi-Banach spaces with a unique unconditional basis, up to permutation. The proof follows the steps of the corresponding uniqueness result for . This will require first to extend some concepts and results from [7] to quasi-Banach sequence spaces. Despite the fact that the changes are minor, we decided to include them here for reference and for future use.

*Definition 2.1. *Borrowing the definition from [7, Section 5], if is a quasi-Banach space with unconditional basis we will say that is *left-dominant with constant * if whenever and are two disjoint sequences in with and such that supp supp for , then . Similarly, we will say that is* right-dominant with constant * if whenever and are two disjoint sequences in with and such that supp supp for , then . We will refer to any normalized unconditional basic sequence as being left-or right-dominant if the the associated sequence space is left-or right-dominant respectively.

For example, the spaces are left- and right-dominant when . In turn, is a right-dominant space as shown in [7,Proposition 5.12], and by the Remark thereafter we infer that so is for .

Our next results, Propositions 2.2 and 2.5, are essentially Theorems 5.6 and 5.7 of [7], respectively, slightly modified to suit our purposes.

Proposition 2.2 (see [7, Theorem 5.6]). *Let be a left- or right-dominant quasi-Banach space with unconditional basis . Suppose that () is a complemented normalized disjoint sequence in then is permutatively equivalent to a subsequence of .*

*Proof. *The proof in the locally convex case applies almost verbatim to this setting and hence we omit it.

The next two lemmas will be used in what follows.

Lemma 2.3 (see [7, Proposition 5.4]). *Suppose that is a left- (resp, right-) dominant basis of a quasi-Banach space and that is a permutation of the natural numbers such that is also left- (resp, right-) dominant. then there is a constant such that for any ,
**
resp,
*

We say that is* equivalent to its square* if is permutatively equivalent to the basis of .

Lemma 2.4 (see [7,Proposition 5.5]). *Let be a left- or right-dominant basis of a quasi-Banach space . In order that be equivalent to its square it is necessary and sufficient that be equivalent to .*

Recall from the theory of Schauder bases that if and are basic sequences in a quasi-Banach space , then * dominates * if for all choices of scalars , whenever then .

Proposition 2.5 (see [7,Theorem 5.7]). *Suppose that is a quasi-Banach space with a left- (or right-) dominant unconditional basis which induces a -convex lattice structure on for some and such that is equivalent to its square. Assume that the Banach envelope of is lattice anti-Euclidean. Then, *(1)*every complemented normalized unconditional basic sequence in is permutatively equivalent to a subsequence of , *(2)* has a unique unconditional basis, up to permutation. *

*Proof. *Consider the left-dominant case. This assumption combined with the fact that is equivalent to its square implies by Lemma 2.4 that and are equivalent. If is a complemented normalized unconditional basic sequence in , then, by Theorem 1.1, is equivalent to a complemented disjoint sequence in , which by Proposition 2.2 will in turn be permutatively equivalent to a subsequence of . Thus (1) follows.

To show (2), suppose that is a normalized unconditional basis of . Applying Theorem 1.1 again we see that is equivalent to a complemented disjoint sequence of the -fold basis of for some . The sequence written in the obvious order is easily seen to be left-dominant, so is permutatively equivalent to a subset of . On the other hand, is permutatively equivalent to a subset of , which is permutatively equivalent to . Theorem 1.2 yields that and are permutatively equivalent.

The sequence dominates by Lemma 2.3, and similarly dominates . Since and are equivalent we deduce that is equivalent to . Now is dominated by and dominates and so is also equivalent to . Therefore is equivalent to . Now is equivalent to and hence to . Since , is equivalent to and the proof is complete.

Theorem 2.6. *If , the -convexified Tsirelson space and all its complemented subspaces with unconditional basis have a unique unconditional basis, up to permutation.*

*Proof. *For , the space is -convex, right-dominant, and its Banach envelope, , is lattice anti-Euclidean. Besides, a straightforward argument on equivalence of basic sequences combined with the fact that and are equivalent in (see [17, page 14]) yields that and are equivalent also in . An appeal to Lemma 2.4 yields the equivalence of and so that is lattice isomorphic to its square. Hence the uniqueness of unconditional basis in is a consequence of Proposition 2.5(2).

If is a (finite or infinite-dimensional) complemented subspace of with normalized unconditional basis , then is permutatively equivalent to a subsequence of by Proposition 2.5(1). Clearly, is right-dominant and equivalent to (see [17, page 14]) and the result follows in the same way as in the preceding paragraph.

*Remark 2.7. *For the -convexification of does not have a unique unconditional basis up to permutation. Indeed, as Kalton pointed out, this is so because can be represented as , and, in this sum, the factor has an unconditional basis containing among its vectors an with (see [18, page 1649]). This also implies that, in this case, is sufficiently Euclidean and so the arguments in the proofs of Theorems 2.6 and [7,Theorem 5.7] will not work.

Our work leaves open the following uniqueness questions.

*Problem 1. *Let . Does the space have unique unconditional basis up to a permutation?

*Problem 2. *It is known that fails to have a unique unconditional basis up to permutation [19]. It would be interesting to know whether the same holds or not in the spaces for .

*Problem 3. *Determine if has a unique unconditional basis up to permutation when and when .

#### Acknowledgments

The first-named author acknowledges the support from the Spanish Ministerio de Ciencia e Innovación Research Project *Operadores, retículos, y geometrtía de espacios de Banach,* Reference no. MTM2008-02652/MTM.