Weakly compact approximation in Banach spaces

The Banach space has the weakly compact approximation property (W.A.P. for short) if there is a constant so that for any weakly compact set and there is a weakly compact operator satisfying and . We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space ) have the W.A.P, but that James' tree space fails to have the W.A.P. It is also shown that the dual has the W.A.P. It follows that the Banach algebras and , consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space so that fails to have the W.A.P., but has this approximation property without the uniform bound .