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28 March 2024
 
  » arxiv » math.OA/9911110

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M-Complete approximate identities in operator spaces
A. Arias ; Haskell P. Rosenthal ;
Date 15 Nov 1999
Subject Operator Algebras; Functional Analysis MSC-class: 47L25, 46B20 (Primary) 46B28, 46L05 (Secondary) | math.OA math.FA
AbstractThis 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 $cal Ksubsetcal Asubset B(ell^2)$. 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 $lambda$-nuclear for some $lambda ge1$; (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.
Source arXiv, math.OA/9911110
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