Uniqueness of unconditional bases in nonlocally convex \(\ell _{1}\)-products
A quasi-Banach space \(X\) with an unconditional basis is said to have a unique unconditional basis up to permutation if for any two normalized unconditional bases of \(X\) there exists an automorphism of \(X\) which takes one basis to a permutation of the other. In [\textit{J.~Bourgain, P. G.~Casazza, J.~Lindenstrauss} and \textit{L.~Tzafriri}, Mem. Am. Math. Soc. 54, 1--111 (1985; Zbl 0575.46011)] it was posed the problem: Does the space \(\ell_1(X)\) have a unique unconditional basis up to permutation whenever the Banach space \(X\) does? In the paper under review, the authors translate this question to the nonlocally convex setting. They prove, in particular, the following corollary: Suppose that \(X\) is a quasi-Banach space with unconditional basis and whose \(q\)-Banach envelope is isomorphic to \(\ell_q\) for some \(0<q<1\). Then the space \(\ell_1(X)\) has a unique unconditional basis up to permutation. The \(q\)-Banach envelope of a quasi-normed space \((X,\|\;\|)\) is the completion of \(X\) in the quasi-norm, generated by the set \(\{\sum_1^na_ix_i: \sum_i|a_i|^q\leq 1\), \(\|x_i\|\leq 1\), \(n\in \mathbb{N}\}\).
uniqueness of unconditional basis up to permutation
scientific article; zbMATH DE number 5817864
scientific article