Unconditional structure of Banach spaces with few operators

TL;DR

This paper constructs specific Banach spaces with unique unconditional bases that contain diverse spreading models, providing counterexamples to longstanding open problems about the structure and classification of such spaces.

Contribution

It introduces a family of Banach spaces with unique unconditional bases containing diverse spreading models, solving a 40-year-old open problem and disproving a conjecture in structure theory.

Findings

Constructed Banach spaces with unique unconditional bases and diverse spreading models.

Provided counterexamples to the classification of spaces with unique unconditional bases.

Disproved the conjecture that spaces with a unique unconditional basis are isomorphic to their square.

Abstract

This article was initially motivated by our goal to show that the Banach space $\mathbb{G}$ constructed by Gowers in [W. T. Gowers, A solution to Banach's hyperplane problem, Bull. London Math. Soc. 26 (1994), no. 6, 523-530] to settle Banach's hyperplane problem has a unique unconditional basis. This uniqueness result served as a springboard to ask whether further structural insights could be derived by rigging Gowers' original construction. As it turned out, the $p$-convexification of $\mathbb{G}$ for $1< p<\infty$, $p\not=2$, provides a family of Banach spaces, each of them with a unique unconditional basis containing block bases whose spreading models are not equivalent to the unit vector basis of $\ell_1$, $\ell_2$, or $c_0$. This solves in the negative a forty-year-old open problem raised by Bourgain et al. in their 1985 \textit{Memoir}, [J. Bourgain, P. G. Casazza, J. Lindenstrauss, and L. Tzafriri, Banach spaces with a unique unconditional basis, up to permutation, Mem. Amer. Math. Soc. 54 (1985), no. 322, iv+111] where they studied the uniqueness of unconditional structure in infinite direct sums of those three spaces with the aim to classify all Banach spaces with a unique unconditional basis. As a by-product of our work, we also disprove the conjecture in structure theory that a space having a unique unconditional basis must be isomorphic to its square, and evince that when a Banach space $\mathbb{X}$ with an unconditional basis has few operators, then the space itself and all its complemented subspaces have a unique unconditional structure.