\(C(K, X)\) as an M-ideal in \(WC(K, X)\) (Q1207756)

A subspace \(J\) of a Banach space \(X\) is said to be an \(M\)-ideal if \(J^ \perp \oplus_ 1N=X^*\) \((\ell_ 1\)-sum) for some closed subspace \(N\). If \(X\) is infinite dimensional and \(K\) is an infinite compact Hausdorff space, then \(WC(K,X)\) (resp. \(C(K,X))\) denotes the Banach space of \(X\)- valued weakly continuous (resp. continuous) functions on \(K\), equipped with supremum norm. The author obtains some positive and negative results about the question: ``When is \(C(K,X)\) an \(M\)-ideal of \(WC(K,X)\)?'' He first disposes off the trivial solutions (i.e. \(WC(K,X)=C(K,X))\).