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We show that every unconditional basis in a finite direct sum$\bigoplus_{p\in A} \ell_p$, with $A\subset (0,\infty]$, splits intounconditional bases of each summand. This settles a 40 year old question raisedin [A. Orty\'nski, 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 $Z=\bigoplus_{p\in A} \ell_p$, $Z\oplus\ell_{2}$, and $Z\oplus c_{0}$ have a unique unconditional basis up topermutation.

Projections and unconditional bases in direct sums of $\ell_p$ spaces, $0