M-IDEALS OF COMPACT-OPERATORS ARE SEPARABLY DETERMINED

We prove that the space K(X) of compact operators on a Banach space X is an M-ideal in the space L(X) of bounded operators if and only if X has the metric compact approximation property (MCAP), and K(Y) is an M -ideal in L(Y) for all separable subspaces Y of X having the MCAP. It follows that the Kalton-Werner theorem characterizing M-ideals of comp act operators on separable Banach spaces is also valid for non-separab le spaces: for a Banach space X, K(X) is an M-ideal in L(X) if and onl y if X has the MCAP, contains no subspace isomorphic to l(1), and has property (M). It also follows that K(Z, X) is an M-ideal in L(Z, X) fo r all Banach spaces Z if and only if X has the MCAP, and K(l(1),X) is an M-ideal in L(l(1),X).