Projections and unconditional bases in direct sums of ℓp spaces, 0<p≤∞

We show that every unconditional basis in a finite direct sum ⨁ p ∈ A ℓ , with ⊂ ( 0 ∞ ] splits into bases of each summand. This settles 40 years old question raised “A. Ortyński, Unconditional ⊕ q < 1 Math. Nachr. 103 (1981), 109–116”. As an application we obtain for any finite, the spaces Z = 2 and c have unique up to permutation.