  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 $|| V|| \leq C$.
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 $C$.
