Weakly Compact Approximation in Banach Spaces

The Banach space E has the weakly compact approximation property (W.A.P. for short) if there is a constant C < ∞ so that for any weakly compact set D ⊂ E and ε > 0 there is a weakly compact operator V : E → E satisfying supx∈D ‖x − V x‖ < ε and ‖V ‖ ≤ C. 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 J) have the W.A.P, but that James’ tree space JT fails to have the W.A.P. It is also shown that the dual J∗ has the W.A.P. It follows that the Banach algebras W (J) and W (J∗), consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space Y so that Y fails to have the W.A.P., but Y has this approximation property without the uniform bound C.