[Authors]
 O. I. Reinov

[Title]
 How bad can a Banach space with the approximation property be? II

[AMS Subj-class]
 46B28 Spaces of operators; tensor products; approximation properties

[Abstract]
 We investigate Banach spaces possessing the Grothendieck
 approximation property which do not have the $C$-metric compact
 (or weakly compact) approximation properties for $1\le C<\infty$.
 As an example let us cite the following result (Corollary 6):
 for each $\alpha\ge1$ there exist separable Banach spaces $E$ and $F$
 and an operator $T:E\to F$ such that $E$ is $\alpha$-complemented
 in $E^{**}$, the space $F$ is reflexive, $T\in\alpha-MAP$,
 but $T\notin\beta-CAP$ if $\beta<\alpha$; moreover, all the duals
 of $E$ -- $E^*, E^{**},...$ -- have the property $MCAP$.

[Comments]
 AMSTeX, 11 pp., Russian

[Contact e-mail]
 orein@orein.usr.pu.ru