M-Complete approximate identities in operator spaces

Abstract

This work introduces the concept of an M-complete approximate identity (M-cai) for a given operator subspace X of an operator space Y. M-cai's generalize central approximate identities in ideals in $C^*$-algebras, for it is proved that if X admits an M-cai in Y, then X is a complete M-ideal in Y. It is proved, using ``special'' M-cai's, that if $\cal J$ is a nuclear ideal in a $C^*$-algebra $\cal A$, then $\cal J$ is completely complemented in Y for any (isomorphically) locally reflexive operator space Y with $\cal J \subset Y \subset \cal A$ and $Y/\cal J$ separable. (This generalizes the previously known special case where $Y=\cal A$, due to Effros-Haagerup.) In turn, this yields a new proof of the Oikhberg-Rosenthal Theorem that $\cal K$ is completely complemented in any separable locally reflexive operator superspace, $\cal K$ the $C^*$-algebra of compact operators on $\ell^2$. M-cai's are also used in obtaining some special affirmative answers to the open problem of whether $\cal K$ is Banach-complemented in $\cal A$ for any separable $C^*$-algebra $\cal A$ with $\cal K\subset\cal A\subset B(\ell^2)$. It is shown that if conversely X is a complete M-ideal in Y, then X admits an M-cai in Y in the following situations: (i) Y has the (Banach) bounded approximation property; (ii) Y is 1-locally reflexive and X is $λ$-nuclear for some $λ\ge1$; (iii) X is a closed 2-sided ideal in an operator algebra Y (via the Effros-Ruan result that then X has a contractive algebraic approximate identity). However it is shown that there exists a separable Banach space X which is an M-ideal in $Y=X^{**}$, yet X admits no M-approximate identity in Y.