Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces

TL;DR

This paper introduces a unified method to analyze the uniqueness of unconditional bases in infinite direct sums of quasi-Banach spaces, demonstrating that many vector-valued sequence spaces have a unique unconditional basis up to permutation.

Contribution

It provides a new approach to establish the uniqueness of unconditional bases in a broad class of quasi-Banach space sums, including spaces with non-locally convex structures.

Findings

Infinite direct sums of certain quasi-Banach spaces have unique unconditional bases.

The method applies to non-locally convex Orlicz and Lorentz sequence spaces.

The approach solves an open problem regarding $ ext{ell}_1(X)$ spaces.

Abstract

This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits us to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from [F. Albiac and C. Ler\'anoz, Uniqueness of unconditional bases in nonlocally convex $\ell_1$-products, J. Math. Anal. Appl. 374 (2011), no. 2, 394--401] we show that if $X$ is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum $\ell_{1}(X)$ has a unique unconditional basis up to a permutation, even without knowing whether $X$ has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.