Weakly compact approximation in Banach spaces

Abstract

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 $\epsilon >0$ there is a weakly compact operator $V:E\to E$ satisfying ${\sup }_{x\in D}||x-Vx||<\epsilon$ and $||V||\le C$. 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 $J^{\ast }$ has the W.A.P. It follows that the Banach algebras $W(J)$ and $W(J^{\ast })$, 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 $C$.

39 pages, plain tex document