An approximation property related to $M$-ideals of compact operators

Abstract:

We investigate a variant of the compact metric approximation property which, for subspaces $X$ of ${c_0}$, is known to be equivalent to $K(X)$, the space of compact operators on $X$, being an $M$-ideal in the space of bounded operators on $X,L(X)$. Among other things, it is shown that an arbitrary Banach space $X$ has this property iff $K(Y,X)$ is an $M$-ideal in $L(Y,X)$ for all Banach spaces $Y$ and, furthermore, that $X$ must contain a copy of ${c_0}$. The proof of the central theorem of this note uses a characterization of those Banach spaces $X$ for which $K(X)$ is an $M$-ideal in $L(X)$ obtained earlier by the second author, as well as some techniques from Banach algebra theory.