Johnson's projection, Kalton's property (M*), and M-ideals of compact operators

Let X and Y be Banach spaces. We give a "non-separable" proof of the Kalton-Werner-Lima-Oja theorem that the subspace K(X, X) of compact operators forms an M-ideal in the space C(X, X) of all continuous linear operators from X to X if and only if X has Kalton's property (M*) and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe how Johnson s projection P on L(X, Y)* applies to f is an element of L(X, Y)* when f is represented via a Borel (with respect to the relative weak* topology) measure on B(X)**circle times B(Y)(w)* subset of L(X, Y)*: f Y* has the Radon-Nikodym property, then P "passes under the integral sign". Our basic theorem en route to this description-a structure theorem for Borel probability measures on B(X)**circle times B(Y)(w)* -also yields a description of K(X, Y)* due to Feder and Saphar. Second, we show that property (M*) for X is equivalent to every functional in B(X)**circle times B(X)*(w)* behaving as if K(X, X) were an M-ideal in L(X, X).