M-complete approximate identities in operator spaces

This work introduces the concept of an M-complete approximate identity (M-c ai) for a given operator subspace X of an operator space Y. M-cai's general ize central approximate identities in ideals in C*-algebras, for it is prov ed 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 J is a nuclear ideal in a C*-alge bra A, then J is completely complemented in Y for any (isomorphically) loca lly reflexive operator space Y with J subset of Y subset of. A and Y/J sepa rable. (This generalizes the previously known special case where Y = A, due to Effros-Haagerup.) In turn, this yields a new proof of the Oikhberg-Rose nthal Theorem that K: is completely complemented in any separable locally r eflexive operator superspace, where K: is the C*-algebra of compact operato rs on l(2). M-cai's are also used in obtaining some special affirmative ans wers to the open problem of whether K: is Banach-complemented in A for any separable C*-algebra A with K subset of A subset of B(l(2)). It is shown th at 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 lambda -nuclear for some lambda greater than or equal to 1; (iii) X is a closed a-sided ideal in an operator algebra Y (via the Effros-Ruan result that then X has a contractiv e 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.