AN IDEAL CHARACTERIZATION OF WHEN A SUBSPACE OF CERTAIN BANACH-SPACESHAS THE METRIC COMPACT APPROXIMATION PROPERTY

C.-M. Cho and W. B. Johnson showed that if a subspace E of l(p), 1 <p < infinity, has the compact approximation property, then K(E) is an M- ideal in L(E). We prove that for every r,s is an element of ]0, 1] wit h r(2) + s(2) < 1, the James space can be provided with an equivalent norm such that an arbitrary subspace E has the metric compact approxim ation property iff there is a norm one projection P on L(E) with Ker P = K(E)(perpendicular to) satisfying //f// greater than or equal to r //Pf// + s//phi - Pf// For All f is an element of L(E). A similar res ult is proved for subspaces of upper p-spaces (e.g. Lorentz sequence s paces d(w,p) and certain renormings of L-p).