1999/11/15 by Alvaro Arias, A. Arias, Arias, A. +2
Mathematics · #46B20 (Primary) 46B28 #46L05 (Secondary) #47L25 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46B20 #msc:46B28 #msc:46L05 #msc:47L25
paper · pdf · doi:10.48550/arxiv.math/9911110
55 pages, AMSTeX, inc. eps figures
arxiv created 1999/11/15 · openalex publication_date 1999/11/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ⊂ Y ⊂ \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 ℓ2. 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⊂\cal A⊂ B(ℓ2). 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 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.