Abstract
AbstractThis 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 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 Albiac and Leránoz (J Math Anal Appl 374(2):394–401, 2011. 10.1016/j.jmaa.2010.09.048) we show that if $$X$$
X
is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum $$\ell _{1}(X)$$
ℓ
1
(
X
)
has a unique unconditional basis up to a permutation, even without knowing whether $$X$$
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.
Funder
ministerio de ciencia, innovación y universidades
Publisher
Springer Science and Business Media LLC
Subject
General Mathematics,Theoretical Computer Science,Analysis
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