On Subspaces of c0 and Extension of Operators into C(K)‐Spaces

Johnson and Zippin recently showed that if X is a weak^*‐closed subspace of ℓ1 and T : X → C(K) is any bounded operator then T can be extended to a bounded operator T̃ : ℓ1 → C(K). We give a converse result: ifX is a subspace of ℓ1 such that ℓ1/X has an unconditional finite‐dimensional decomposition (UFDD) and every operator T : X → C(K) can be extended to ℓ1 then there is an automorphism τ of ℓ1 such that τ(X) is weak^*‐closed. This result is proved by studying subspaces of c0 and several different characterizations of such subspaces are given.