m at h . O A ] 1 5 N ov 1 99 9 M - COMPLETE APPROXIMATE IDENTITIES IN OPERATOR SPACES
نویسندگان
چکیده
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 J is a nuclear ideal in a C *-algebra A, then J is completely complemented in Y for any (isomorphically) locally reflexive operator space Y with J ⊂ Y ⊂ A and Y /J separable. (This generalizes the previously known special case where Y = A, due to Effros-Haagerup.) In turn, this yields a new proof of the Oikhberg-Rosenthal Theorem that K is completely complemented in any separable locally reflexive operator su-perspace, K the C *-algebra of compact operators on ℓ 2. M-cai's are also used in obtaining some special affirmative answers to the open problem of whether K is Banach-complemented in A for any separable C *-algebra A with K ⊂ A ⊂ B(ℓ 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 ≥ 1; (iii) X is a closed 2-sided ideal in an operator algebra Y (via the Effros-Ruan result that then X has a con-tractive 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 .
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