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