Unconditional structure of Banach spaces with few operators

This paper demonstrates that the $p$-convexification of Gowers' space $\mathbb{G}$ yields Banach spaces with unique unconditional bases containing block bases with spreading models distinct from $\ell_1$, $\ell_2$, and $c_0$, thereby resolving a long-standing open problem by Bourgain et al. and disproving the conjecture that such spaces must be isomorphic to their squares.

The Gist

Here is an explanation of the paper "Unconditional Structure of Banach Spaces with Few Operators" by F. Albiac and J. L. Ansorena, translated into everyday language with analogies.

The Big Picture: The "Lego" Problem

Imagine you are building with Banach Spaces. In the world of mathematics, these are complex, infinite-dimensional structures (like infinite Lego towers). To understand these towers, mathematicians use bases. Think of a basis as the specific set of Lego bricks you use to build the tower.

Usually, you can build the same tower using different sets of bricks. For example, you could build a wall using red bricks or blue bricks; the wall looks the same, but the "basis" (the bricks) is different.

However, some very special towers have a Unique Unconditional Basis. This means that no matter how you try to rearrange or swap out the bricks, you must use that exact specific set of bricks to build the tower. If you try to use a different set, the tower collapses or changes shape.

For a long time, mathematicians thought these "Unique Brick Towers" were very boring and predictable. They believed:

1. They all looked like their own "squares" (if you put two identical towers side-by-side, it looks just like one big tower).
2. Their internal structure was always one of three simple patterns: like a line ($\ell_1$), a plane ($\ell_2$), or a stack of boxes ($c_0$).

The Goal: Breaking the Rules

The authors of this paper wanted to test these beliefs. They asked:

• Question A: Can we build a Unique Brick Tower that doesn't look like its own square? (i.e., Two of them side-by-side look totally different from one).
• Question B: Can we build a Unique Brick Tower that has a weird internal structure, something that isn't just a line, a plane, or a stack of boxes?

The Solution: The "Gowers" Machine

To solve this, the authors used a famous, complicated machine built by a mathematician named Gowers (let's call it the Gowers Machine). This machine was originally built to solve a different puzzle about "hyperplanes" (slicing the tower in half).

The authors took this Gowers Machine and applied a mathematical process called p-convexification.

• The Analogy: Imagine the Gowers Machine is a block of clay. "p-convexification" is like baking that clay at different temperatures.
  • If you bake it at temperature $p=2$, it becomes hard and rigid (like standard $\ell_2$).
  • If you bake it at temperature $p=3$ or $p=1.5$, it changes its texture completely. It becomes a new, strange material.

The authors proved that if you bake this Gowers Machine at almost any temperature (except 2), you get a new type of Unique Brick Tower.

The Surprising Discoveries

These new towers broke all the old rules:

1. They are not "Self-Similar":
   If you take two of these new towers and put them together, the result is not the same as the original tower.

   • Analogy: Imagine a puzzle piece. Usually, if you put two identical puzzle pieces together, they form a bigger version of the same shape. But these new pieces are weird: if you put two together, they form a shape that looks nothing like the original. This disproved a 40-year-old guess that all Unique Brick Towers must be "self-similar."

2. They have "Exotic" Internal Structures:
   The internal "spreading models" (how the bricks stretch out over infinity) of these towers are not the standard line, plane, or stack.

   • Analogy: For 40 years, mathematicians thought all Unique Brick Towers were made of standard Lego bricks. The authors found towers made of "alien" bricks that don't fit into any standard category. This answered a famous question by Bourgain, Casazza, Lindenstrauss, and Tzafriri with a big "No."

3. They have "Few Operators":
   This is the secret sauce. In math, an "operator" is a way to transform the tower (stretch it, rotate it, squish it).

   • Most towers have infinite ways to transform them.
   • The Gowers Machine (and its baked versions) has very few ways to transform it. Almost any transformation you try to do is either just a simple diagonal adjustment or it destroys the structure (strictly singular).
   • Because there are so few ways to mess with the tower, the tower is forced to keep its unique identity. It's like a fortress with only one gate; if you try to enter through the gate, you have to follow the rules, so the structure remains unique.

The "Few Operators" Rule

The paper also discovered a general rule: If a tower has very few ways to transform it (few operators), then every part of that tower that can be separated out (complemented subspaces) also has a unique structure.

It's like saying: "If a company has a very strict, unique culture and very few ways to reorganize its departments, then every single department inside that company will also have a unique, unchangeable culture."

Why Does This Matter?

Before this paper, mathematicians were trying to classify all these "Unique Brick Towers" and thought the list was short and predictable.

• The Result: The authors showed the list is actually much longer and weirder than we thought.
• The Impact: They proved that the "Unique Structure" property doesn't force a space to be "self-similar" (isomorphic to its square). They found a whole new family of spaces that are rigid and unique, but also strange and non-standard.

Summary in One Sentence

The authors took a weird, rigid mathematical machine (Gowers space), baked it at different temperatures, and discovered a whole new family of mathematical structures that are unique and rigid, yet completely different from anything we thought was possible, proving that the universe of Banach spaces is far more exotic than we imagined.

Technical Summary

Here is a detailed technical summary of the paper "Unconditional Structure of Banach Spaces with Few Operators" by F. Albiac and J. L. Ansorena.

1. Problem Statement and Motivation

The paper addresses several long-standing open problems in the structure theory of Banach spaces, specifically concerning the uniqueness of unconditional bases.

• Context: Historically, it was believed that Banach spaces with a unique unconditional basis (up to equivalence and permutation) were rare and structurally simple (e.g., $c_0, \ell_1, \ell_2$). A prevailing conjecture suggested that if a space has a unique unconditional basis, it must be isomorphic to its square ($X \cong X^2$) and possess "nice" spreading models (equivalent to $\ell_1, \ell_2,$ or $c_0$).
• Specific Open Questions:
  1. Question 2.2: Is there a Banach space with a unique unconditional basis that is not isomorphic to its square? (i.e., fails to be "power stable").
  2. Question 2.3: Does there exist a space with a unique unconditional basis where the optimal lattice convexity/concavity indices $\sigma(K)$ and $\tau(K)$ lie outside the set $\{1, 2, \infty\}$?
  3. Question 2.7: Can $\ell_p$ (for $p \notin \{1, 2\}$) be crudely finitely complementably represented in a space with a unique unconditional basis?
  4. Question 2.8 (Bourgain et al., 1985): Is every unconditional spreading model of a space with a unique unconditional basis equivalent to the unit vector basis of $\ell_1, \ell_2,$ or $c_0$?

The authors aim to resolve these questions by analyzing the Gowers space ($G$), originally constructed to solve Banach's hyperplane problem, and its $p$-convexifications.

2. Methodology and Framework

The authors employ a combination of operator theory, lattice theory, and combinatorial arguments specific to spaces with "few operators."

• The D+S Property (Diagonal + Strictly Singular):
  The core methodological tool is the study of Banach spaces where every bounded linear operator $T$ can be decomposed as $T = D + S$, where $D$ is a diagonal operator relative to the basis and $S$ is strictly singular. This property was established for Gowers space $G$ by Gowers and Maurey. The authors generalize this to a class of bases called D+S unconditional bases.
• Operator Decomposition:
  They utilize the fact that in spaces with the D+S property, the space of operators is "small." This allows them to prove strong rigidity results:
  • Any complemented subspace is congruent to a block subspace $[X|A]$.
  • Any two subbases are permutatively equivalent only if they differ by a finite set.
  • The space is not isomorphic to any of its proper subspaces.
• $p$-Convexification:
  The authors construct new spaces $G[p, f]$ by applying $p$-convexification to the Gowers space $G[f]$ (where $f$ is a function from Schlumprecht's class $\mathcal{F}$). This operation alters the linear structure and the lattice indices ($\sigma$ and $\tau$) while preserving the "few operators" property.
• Spreading Models:
  They analyze the asymptotic behavior of block sequences to determine spreading models, utilizing Krivine's theorem and estimates on disjointly supported sequences.

3. Key Contributions and Results

The paper provides a unified framework to generate a family of counterexamples to the classical structural conjectures.

A. The D+S Property for Convexified Gowers Spaces

Theorem 5.5: The authors prove that the canonical unconditional basis of the $p$-convexified Gowers space $G[p, f]$ (for $1 \le p < \infty$) possesses the D+S property.

• Significance: This extends the "few operators" phenomenon from the original Gowers space to a whole family of spaces with different convexity indices.

B. Uniqueness of Unconditional Basis and Structural Rigidity

Theorem 5.6 (Main Result): Let $X$ be an infinite-dimensional complemented subspace of $G[p, f]$. Then:

1. Uniqueness: $X$ has a unique unconditional basis.
2. Failure of Power Stability: $X$ is not isomorphic to its square ($X \not\cong X^2$). This provides a negative answer to Question 2.2.
3. Failure of Hyperplane Stability: $X$ is not hyperplane-sum stable.
4. Lattice Indices: $\sigma(X) = p$. This answers Question 2.3 affirmatively for $p \in (1, 2) \cup (2, \infty)$, showing that unique unconditional bases can exist outside the classical indices $\{1, 2, \infty\}$.
5. Exotic Spreading Models: The spreading models of $X$ are not equivalent to $\ell_1, \ell_2,$ or $c_0$ (unless $p=1$ or $p=2$). Specifically, $\ell_p$ appears as a spreading model. This provides a negative answer to Question 2.8.

C. General Structural Theorems for D+S Spaces

The authors establish general theorems for any Banach space with a D+S unconditional basis:

• Theorem 4.10: Every complemented subspace has a unique unconditional basis.
• Theorem 4.11: If two block subspaces $[X|M]$ and $[X|K]$ are isomorphic, then $|M \setminus K| = |K \setminus M| < \infty$. This implies a lack of self-similarity.
• Corollary 4.12: Such spaces cannot contain complemented copies of spaces that are power stable or hyperplane stable (like $\ell_p$ for $p \neq 2$).

4. Significance and Impact

This paper fundamentally shifts the understanding of Banach spaces with unique unconditional bases:

1. Disproving the "Square" Conjecture: It definitively shows that having a unique unconditional basis does not imply that the space is isomorphic to its square. This breaks a long-held heuristic in the field.
2. Classification of Spreading Models: It refutes the conjecture that unique unconditional bases must have "classical" spreading models ($\ell_1, \ell_2, c_0$). The authors demonstrate that spaces with unique bases can have spreading models isomorphic to $\ell_p$ for any $p \in (1, \infty)$.
3. Richness of the Class: The work reveals that the class of spaces with unique unconditional bases is much richer and more diverse than previously thought, encompassing spaces with arbitrary convexity indices and complex asymptotic structures.
4. Methodological Advancement: The introduction and rigorous application of the D+S property as a tool to prove uniqueness and structural rigidity provides a powerful new toolkit for future research in Banach space theory.

5. Open Questions Remaining

The paper concludes by identifying new frontiers:

• Question 6.2: Does the direct sum $G^m$ (for $m \ge 2$) have a unique unconditional basis?
• Question 6.3: Does the $p$-convexification $G(p)$ for quasi-Banach spaces ($0 < p < 1$) have a unique unconditional basis? The authors note that current proofs fail in this regime due to the lack of local convexity and the failure of the basis to be shrinking.

In summary, Albiac and Ansorena successfully use the rigidity of Gowers-type spaces to construct a family of Banach spaces that simultaneously possess a unique unconditional basis while exhibiting "exotic" structural properties (non-self-similarity, non-classical spreading models), thereby resolving several decades-old open problems in the negative.