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## Studia Mathematica

2000 | 141 | 2 | 143-200
Tytuł artykułu

### M-complete approximate identities in operator spaces

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EN
Abstrakty
EN
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 superspace, where K is 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 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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
143-200
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-11-23
poprawiono
2000-03-14
Twórcy
autor
• Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249-0664, U.S.A.
autor
• Department of Mathematics, University of Texas at Austin, Austin, TX 78712-1082, U.S.A.
Bibliografia
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