An Approximation Property Related to M -ideals of Compact Operators

We investigate a variant of the compact metric approximation property which, for subspaces X of c0 , is known to be equivalent to K(X), the space of compact operators on X , being an A/-ideal in the space of bounded operators on X , L{X). Among other things, it is shown that an arbitrary Banach space X has this property iff K(Y, X) is an Af-ideal in L(Y, X) for all Banach spaces Y and, furthermore, that X must contain a copy of c0 . The proof of the central theorem of this note uses a characterization of those Banach spaces X for which K(X) is an Af-ideal in L(X) obtained earlier by the second author, as well as some techniques from Banach algebra theory. A closed subspace / of a Banach space X is called an Af-ideal iff the dual space X* is the /,-direct sum of the annihilator J and some closed subspace, a notion that has been introduced by Alfsen and Effros in order to unify certain aspects of the theory of C*-algebras and A(K)-spaces [1]. Though not using this terminology, Dixmier had already proved in 1950 [6] that, for any Hubert space H, K(H) is an Af-ideal in L(H), and since the early 70s the question of which Banach spaces X the space K(X) is an Af-ideal for in L(X) has been of general interest for various reasons (for a brief survey on the connections to the theory of Banach spaces see, e.g., the introduction to [12] or [9], and for some applications consult the appropriate references in [13]). So it was shown, for example in [8], that X with K(X) being an Af-ideal of L(X) necessarily must possess the compact metric approximation property (CMAP), and in [4] it turned out that for subspaces of / , I < p < oo, the CMAP (here, one can dispose of the letter "Af ") already ensures that property. As a matter of fact [13], K(X) is an Af-ideal in L(X) iff X satisfies the following strong version of the CMAP: There is a net (Ka) in the unit ball BK,X) of K(X) converging to the identity strongly and satisfying (A) lim sup \\KaTx + (Id -Ka)T2|| <1 a