Projections and unconditional bases in direct sums of ${\ell }_p$ spaces, $0<p\le \infty$

Abstract

We show that every unconditional basis in a finite direct sum ${⨁}_{p\in A}{\ell }_p$, with $A\subset (0,\infty ]$, splits into unconditional bases of each summand. This settles a 40 year old question raised in [A. Ortyński, Unconditional bases in ${\ell }_p\oplus {\ell }_q,$ $0<p<q<1$, Math. Nachr. 103 (1981), 109-116]. As an application we obtain that for any $A\subset (0,1]$ finite, the spaces $Z={⨁}_{p\in A}{\ell }_p$, $Z\oplus {\ell }_2$, and $Z\oplus c_0$ have a unique unconditional basis up to permutation.