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 $C < \infty$so that for any weakly compact set $D \subset E$ and $\varepsilon > 0$there is a weakly compact operator $V: E \to E$ satisfying $\sup _{x\in D} \Vert x - Vx \Vert < \varepsilon$ and $\Vert V\Vert \leq C$. 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 $C$.