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

Abstract This paper is devoted to providing a unifying approach the study of uniqueness unconditional bases, up equivalence and permutation, infinite direct sums quasi-Banach spaces. Our new this type problem permits show that wide class vector-valued sequence spaces have unique basis permutation. In particular, solving from Albiac Leránoz (J Math Anal Appl 374(2):394–401, 2011. 10.1016/j.jmaa.2010.09.048 ) we if $$X$$ X space with strongly absolute then sum $$\ell _{1}(X)$$ ℓ 1 ( ) has even without knowing whether or not. Applications structure non-locally convex Orlicz Lorentz spaces, among other classical are also obtained as by-product our work.