On subspaces of c_0 and extension of operators into C(K)-spaces

Johnson and Zippin recently showed that if $X$ is a weak^*-closed subspace of $\ell_1$ and T:X-> C(K) is any bounded operator then T can extended to a bounded operator $\tilde T:\ell_1\to C(K).$ We give a converse result: if X is a subspace of $\ell_1$ so that $\ell_1/X$ has a (UFDD) and every operator T:X -> C(K) can be extended to $\ell_1$ then there is an automorphism $\tau$ of $\ell_1$ so that $\tau(X)$ is weak^*-closed. This result is proved by studying subspaces of c_0 and several different characterizations of such subspaces are given.