The Tsirelson Space 𝒯^(𝑝) Has a Unique Unconditional Basis up to Permutation for 0<𝑝<1

Abstract

We show that the 𝑝-convexified Tsirelson space 𝒯^(𝑝) for 0<𝑝<1 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 (𝑋,‖⋅‖) is a quasi-Banach space (in particular a Banach space) with a normalized unconditional basis (𝑒_𝑛)^∞_𝑛=1 (i.e., ‖𝑒_𝑛‖=1 for all 𝑛∈ℕ), 𝑋 is said to have a unique unconditional basis (up to permutation) if whenever (𝑥_𝑛)^∞_𝑛=1 is another normalized unconditional basis of 𝑋, then (𝑥_𝑛)^∞_𝑛=1 is equivalent to (𝑒_𝑛)^∞_𝑛=1 (after a permutation); that is, there exists an automorphism of 𝑋 which takes one basis to (a permutation of) the other.

The problem of uniqueness of unconditional basis is classical. It is well known that 𝑐₀, ℓ₁, and ℓ₂ 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 𝑐₀, ℓ₁, and ℓ₂ 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 𝑐₀(ℓ₁), 𝑐₀(ℓ₂), ℓ₁(𝑐₀), and ℓ₁(ℓ₂) all have unique unconditional bases up to permutation, while, surprisingly, ℓ₂(ℓ₁) and ℓ₂(𝑐₀) 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 𝒯, 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 ℓ_𝑝 spaces for 0<𝑝<1, have a unique unconditional basis. The same is true in nonlocally convex Lorentz sequence spaces [9, 10] and in the Hardy spaces 𝐻_𝑝(𝕋) for 0<𝑝<1 [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, ℓ_𝑝(ℓ_𝑞), ℓ_𝑝(𝑐₀), ℓ_𝑝(ℓ₁), ℓ_𝑝(ℓ₂), 𝑐₀(ℓ_𝑝), ℓ₁(ℓ_𝑝), and ℓ₂(ℓ_𝑝), where 0<𝑝,𝑞<1. With the exception of ℓ₂(ℓ_𝑝) 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 𝑀 so that for any 𝑛∈ℕ there are operators 𝑆_𝑛∶𝑋→ℓ^𝑛₂ and 𝑇_𝑛∶ℓ^𝑛₂→𝑋 so that 𝑆_𝑛∘𝑇_𝑛=𝐼_ℓ^𝑛₂, ‖𝑆_𝑛‖‖𝑇_𝑛‖≤𝑀, and 𝑆_𝑛 is a lattice homomorphism. This is equivalent to asking that ℓ₂ is finitely representable as a complemented sublattice of 𝑋. We will say that 𝑋 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 1-unconditional basis.

Theorem 1.1 (see [14, Theorem 3.4]). Let 𝑌 and 𝑍 be quasi-Banach sequence spaces. Suppose that 𝑍 is 𝑝-convex for some 𝑝>0 and that 𝑌 is isomorphic to a complemented subspace of 𝑍. Suppose that the Banach envelope 𝑌 of 𝑌 is lattice anti-Euclidean. Then there exists 𝑁∈ℕ and a complemented disjoint positive sequence (𝑣_𝑛) in 𝑍^𝑁 that is equivalent to the unit vector basis (𝑢_𝑛) in 𝑌. Furthermore, the projection 𝑃 of 𝑍^𝑁 onto [𝑣_𝑛] may be given in the form 𝑃(𝑧)=^∞_𝑛=1𝑣^∗_𝑛(𝑧)𝑣_𝑛,(1.1) where 𝑣^∗_𝑛≥0 and supp(𝑣^∗_𝑛)⊆supp(𝑣_𝑛) for all 𝑛.

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 (𝑢_𝑛)^∞_𝑛=1 and (𝑣_𝑛)^∞_𝑛=1 are two unconditional basic sequences of a quasi-Banach space 𝑋, then (𝑢_𝑛) and (𝑣_𝑛) are equivalent (up to permutation) if and only if (𝑢_𝑛) is equivalent (up to permutation) to a subsequence of (𝑣_𝑛) and (𝑣_𝑛) is equivalent (up to permutation) to a subsequence of (𝑢_𝑛).

2. Main Result

In this section we settle the question of uniqueness of unconditional basis in the spaces 𝒯^(𝑝) for 0<𝑝<1. These spaces are the nonlocally convex counterpart to the 𝑝-convexification of Tsirelson space for 𝑝>1, 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 0<𝑝<∞, the 𝑝-convexification 𝒯^(𝑝) of Tsirelson space 𝒯 is obtained from 𝒯 by putting

‖𝑥‖_𝒯^(𝑝)=‖(|𝑎_𝑛|^𝑝)^∞_𝑛=1‖_𝒯^1/𝑝(2.1) for those sequences of real numbers 𝑥=(𝑎_𝑛)^∞_𝑛=1 such that (|𝑎_𝑛|^𝑝)^∞_𝑛=1∈𝒯. Equation (2.1) defines a norm for 1≤𝑝 and a 𝑝-norm when 0<𝑝<1. Obviously, the space (𝒯⁽¹⁾,‖⋅‖_𝒯⁽¹⁾) is simply (𝒯,‖⋅‖_𝒯).

The canonical unit vectors form a 1-unconditional basis of 𝒯^(𝑝) for all 0<𝑝<∞. Casazza and Kalton established in [7] the uniqueness of unconditional basis up to permutation of 𝒯 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 2-convexification 𝒯⁽²⁾ of 𝒯.

Here we show that for 𝑝<1, the space 𝒯^(𝑝) 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 𝒯. 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 𝑋 is a quasi-Banach space with unconditional basis we will say that 𝑋 is left-dominant with constant 𝛾≥1 if whenever (𝑢₁,𝑢₂,…,𝑢_𝑛) and (𝑣₁,𝑣₂,…,𝑣_𝑛) are two disjoint sequences in 𝑐₀₀ with ‖𝑢_𝑘‖_𝑋≥‖𝑣_𝑘‖_𝑋 and such that max{supp 𝑢_𝑘}<𝑚𝑖𝑛{supp 𝑣_𝑘} for 1≤𝑘≤𝑛, then ‖∑^𝑛_𝑘=1𝑣_𝑘‖_𝑋≤𝛾‖∑^𝑛_𝑘=1𝑢_𝑘‖_𝑋. Similarly, we will say that 𝑋 is right-dominant with constant 𝛾 if whenever (𝑢₁,𝑢₂,…,𝑢_𝑛) and (𝑣₁,𝑣₂,…,𝑣_𝑛) are two disjoint sequences in 𝑐₀₀ with ‖𝑢_𝑘‖_𝑋≤‖𝑣_𝑘‖_𝑋 and such that max{supp 𝑢_𝑘}<𝑚𝑖𝑛{supp 𝑣_𝑘} for 1≤𝑘≤𝑛, then ‖∑^𝑛_𝑘=1𝑢_𝑘‖_𝑋≤𝛾‖∑^𝑛_𝑘=1𝑣_𝑘‖_𝑋. 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 ℓ_𝑝 spaces are left- and right-dominant when 0<𝑝<∞. In turn, 𝒯 is a right-dominant space as shown in [7,Proposition 5.12], and by the Remark thereafter we infer that so is 𝒯^(𝑝) for 0<𝑝<∞.
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 𝑋 be a left- or right-dominant quasi-Banach space 𝑋 with unconditional basis (𝑒_𝑛)^∞_𝑛=1. Suppose that (𝑢_𝑛)_𝑛∈𝑆 (|𝑆|≤∞) is a complemented normalized disjoint sequence in 𝑋, then (𝑢_𝑛)_𝑛∈𝑆 is permutatively equivalent to a subsequence (𝑒_𝑘𝑛)^∞_𝑛=1 of (𝑒_𝑛)^∞_𝑛=1.

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 (𝑢_𝑛)^∞_𝑛=1 is a left- (resp, right-) dominant basis of a quasi-Banach space 𝑋 and that 𝜋 is a permutation of the natural numbers such that (𝑢_𝜋(𝑛))^∞_𝑛=1 is also left- (resp, right-) dominant. then there is a constant 𝐶 such that for any (𝛼_𝑘)∈𝑐₀₀, ‖^∞_𝑘=1𝛼_𝑘𝑢_2𝑘‖≤𝐶‖^∞_𝑘=1𝛼_𝑘𝑢_𝜋(𝑘)‖(2.2) resp, ‖^∞_𝑘=1𝛼_𝑘𝑢_2𝑘‖≥𝐶⁻¹‖^∞_𝑘=1𝛼_𝑘𝑢_𝜋(𝑘)‖.(2.3)

We say that (𝑢_𝑛) is equivalent to its square if (𝑢_𝑛) is permutatively equivalent to the basis {(𝑢₁,0),(0,𝑢₁),(𝑢₂,0),…,} of [𝑢_𝑛]⊕[𝑢_𝑛].

Lemma 2.4 (see [7,Proposition 5.5]). Let (𝑢_𝑛)^∞_𝑛=1 be a left- or right-dominant basis of a quasi-Banach space 𝑋. In order that (𝑢_𝑛)^∞_𝑛=1 be equivalent to its square it is necessary and sufficient that (𝑢_𝑛)^∞_𝑛=1 be equivalent to (𝑢_2𝑛)^∞_𝑛=1.

Recall from the theory of Schauder bases that if (𝑥_𝑛)^∞_𝑛=1 and (𝑦_𝑛)^∞_𝑛=1 are basic sequences in a quasi-Banach space 𝑋, then (𝑥_𝑛)^∞_𝑛=1 dominates (𝑦_𝑛)^∞_𝑛=1 if for all choices of scalars (𝑎_𝑛)^∞_𝑛=1, whenever ∑^∞_𝑛=1𝑎_𝑛𝑥_𝑛∈[𝑥_𝑛]^∞_𝑛=1 then ∑^∞_𝑛=1𝑎_𝑛𝑦_𝑛∈[𝑦_𝑛]^∞_𝑛=1.

Proposition 2.5 (see [7,Theorem 5.7]). Suppose that 𝑋 is a quasi-Banach space with a left- (or right-) dominant unconditional basis (𝑒_𝑛)^∞_𝑛=1 which induces a 𝑝-convex lattice structure on 𝑋 for some 0<𝑝<1 and such that (𝑒_𝑛)^∞_𝑛=1 is equivalent to its square. Assume that the Banach envelope 𝑋 of 𝑋 is lattice anti-Euclidean. Then, (1)every complemented normalized unconditional basic sequence (𝑢_𝑛)_𝑛∈𝑆 in 𝑋 is permutatively equivalent to a subsequence (𝑒_𝑘𝑛)^∞_𝑛=1 of (𝑒_𝑛)^∞_𝑛=1, (2)𝑋 has a unique unconditional basis, up to permutation.

Proof. Consider the left-dominant case. This assumption combined with the fact that (𝑒_𝑛)^∞_𝑛=1 is equivalent to its square implies by Lemma 2.4 that (𝑒_𝑛)^∞_𝑛=1 and (𝑒_2𝑛)^∞_𝑛=1 are equivalent. If (𝑢_𝑛)_𝑛∈𝑆 is a complemented normalized unconditional basic sequence in 𝑋, then, by Theorem 1.1, (𝑢_𝑛)_𝑛∈𝑆 is equivalent to a complemented disjoint sequence in 𝑋, which by Proposition 2.2 will in turn be permutatively equivalent to a subsequence (𝑒_𝑘𝑛)^∞_𝑛=1 of (𝑒_𝑛)^∞_𝑛=1. Thus (1) follows.
To show (2), suppose that (𝑢_𝑛)^∞_𝑛=1 is a normalized unconditional basis of 𝑋. Applying Theorem 1.1 again we see that (𝑒_𝑛)^∞_𝑛=1 is equivalent to a complemented disjoint sequence of the 𝑁-fold basis (𝑢_𝑛)^𝑁 of 𝑋^𝑁 for some 𝑁∈ℕ. The sequence (𝑓_𝑛)=(𝑒_𝑘𝑛)^𝑁 written in the obvious order is easily seen to be left-dominant, so (𝑒_𝑛)^∞_𝑛=1 is permutatively equivalent to a subset of (𝑒_𝑘𝑛)^𝑁. On the other hand, (𝑒_𝑘𝑛)^𝑁 is permutatively equivalent to a subset of (𝑒_𝑛)^𝑁, which is permutatively equivalent to (𝑒_𝑛). Theorem 1.2 yields that (𝑒_𝑘𝑛)^𝑁 and (𝑒_𝑛) are permutatively equivalent.
The sequence (𝑓_𝑛) dominates (𝑒_2𝑛) by Lemma 2.3, and similarly (𝑒_𝑛) dominates (𝑓_2𝑛). Since (𝑒_𝑛) and (𝑒_4𝑛) are equivalent we deduce that (𝑓_2𝑛) is equivalent to (𝑒_𝑛). Now (𝑓_2𝑛−1)^∞_𝑛=1 is dominated by (𝑓₁,𝑓₂,𝑓₄,…,𝑓_2𝑛−2,…) and dominates (𝑓₁,𝑓₄,𝑓₈,…,𝑓_4𝑛−4,…) and so is also equivalent to (𝑓_2𝑛). Therefore (𝑓_𝑛) is equivalent to (𝑒_𝑛). Now (𝑓_𝑁𝑛) is equivalent to (𝑒_𝑁𝑛) and hence to (𝑒_𝑛). Since 𝑓_𝑁𝑛=(0,…,0,𝑒_𝑘𝑛), (𝑒_𝑘𝑛) is equivalent to (𝑒_𝑛) and the proof is complete.

Theorem 2.6. If 0<𝑝<1, the 𝑝-convexified Tsirelson space 𝒯^(𝑝) and all its complemented subspaces with unconditional basis have a unique unconditional basis, up to permutation.

Proof. For 0<𝑝<1, the space 𝒯^(𝑝) is 𝑝-convex, right-dominant, and its Banach envelope, ℓ₁, is lattice anti-Euclidean. Besides, a straightforward argument on equivalence of basic sequences combined with the fact that (𝑒_𝑛)^∞_𝑛=1 and (𝑒_2𝑛)^∞_𝑛=1 are equivalent in 𝒯 (see [17, page 14]) yields that (𝑒_𝑛)^∞_𝑛=1 and (𝑒_2𝑛)^∞_𝑛=1 are equivalent also in 𝒯^(𝑝). An appeal to Lemma 2.4 yields the equivalence of (𝑒_𝑛) and (𝑒_𝑛)² so that 𝒯^(𝑝) is lattice isomorphic to its square. Hence the uniqueness of unconditional basis in 𝒯^(𝑝) is a consequence of Proposition 2.5(2).
If 𝑌 is a (finite or infinite-dimensional) complemented subspace of 𝒯^(𝑝) with normalized unconditional basis (𝑢_𝑛)_𝑛∈𝑆, then (𝑢_𝑛)_𝑛∈𝑆 is permutatively equivalent to a subsequence (𝑒_𝑘𝑛)^∞_𝑛=1 of (𝑒_𝑛)^∞_𝑛=1 by Proposition 2.5(1). Clearly, (𝑒_𝑘𝑛)^∞_𝑛=1 is right-dominant and equivalent to (𝑒_𝑘2𝑛)^∞_𝑛=1 (see [17, page 14]) and the result follows in the same way as in the preceding paragraph.

Remark 2.7. For 1<𝑝≠2 the 𝑝-convexification 𝒯^(𝑝) of 𝒯 does not have a unique unconditional basis up to permutation. Indeed, as Kalton pointed out, this is so because 𝒯^(𝑝) can be represented as (ℓ¹_𝑝⊕ℓ²_𝑝⊕⋯ℓ^𝑛_𝑝⊕⋯)_𝒯, and, in this sum, the factor ℓ^𝑛_𝑝 has an unconditional basis containing among its vectors an ℓ^𝑘₂ with 𝑘∼log𝑛 (see [18, page 1649]). This also implies that, in this case, 𝒯^(𝑝) 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 0<𝑝<1. Does the space ℓ_𝑝(𝒯) have unique unconditional basis up to a permutation?

Problem 2. It is known that 𝑐₀(𝒯) 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 𝑐₀(𝒯^(𝑝)) for 0<𝑝<1.

Problem 3. Determine if ℓ₁(𝒯^(𝑝)) has a unique unconditional basis up to permutation when 𝑝<1 and when 𝑝=1.

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.