M-complete approximate identities in operator spaces

A. Arias; H. Rosenthal

Studia Mathematica (2000)

  • Volume: 141, Issue: 2, page 143-200
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Arias, A., and Rosenthal, H.. "M-complete approximate identities in operator spaces." Studia Mathematica 141.2 (2000): 143-200. <http://eudml.org/doc/216777>.

@article{Arias2000,
abstract = {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.},
author = {Arias, A., Rosenthal, H.},
journal = {Studia Mathematica},
keywords = {M-complete approximate identity; M-ideal; operator space; nuclear ideal; -algebras; Oikhberg-Rosenthal theorem},
language = {eng},
number = {2},
pages = {143-200},
title = {M-complete approximate identities in operator spaces},
url = {http://eudml.org/doc/216777},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Arias, A.
AU - Rosenthal, H.
TI - M-complete approximate identities in operator spaces
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 2
SP - 143
EP - 200
AB - 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.
LA - eng
KW - M-complete approximate identity; M-ideal; operator space; nuclear ideal; -algebras; Oikhberg-Rosenthal theorem
UR - http://eudml.org/doc/216777
ER -

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