Abstract
Let fRng 1=1 be a commuting approximating sequence of the Banach space X leaving the closed subspace A X invariant. Then we prove three-space results of the following kind: If the operators Rn induce basis projections on X=A, and X or A is anLp-space, then both X and A have bases. We apply these results to show that the spaces C = spanfz k : k2 g C(T) and L = spanfz k : k2 g L1(T) have bases whenever Z and Zn is a Sidon set.
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