SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS

doi:10.1007/s10473-021-0616-0

摘要/Abstract

摘要： We show that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in the following sense: there is a factorization of the identity of $M$ $$Id_M=vu:M{\buildrel u\over\longrightarrow} B(H) {\buildrel v\over\longrightarrow} M$$ with $u$ normal, unital, positive and $v$ completely contractive. As a corollary, if $M$ has a separable predual, $M$ is isomorphic (as a Banach space) to $B(\ell_2)$. For instance this applies (rather surprisingly) to the von Neumann algebra of any free group. Nevertheless, since $B(H)$ fails the approximation property (due to Szankowski) there are $M$'s (namely $B(H)^{**}$ and certain finite examples defined using ultraproducts) that are not seemingly injective. Moreover, for $M$ to be seemingly injective it suffices to have the above factorization of $Id_M$ through $B(H)$ with $u,v$ positive (and $u$ still normal).

关键词: von Neumann algebra, injectivity, positive approximation property

Abstract: We show that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in the following sense: there is a factorization of the identity of $M$ $$Id_M=vu:M{\buildrel u\over\longrightarrow} B(H) {\buildrel v\over\longrightarrow} M$$ with $u$ normal, unital, positive and $v$ completely contractive. As a corollary, if $M$ has a separable predual, $M$ is isomorphic (as a Banach space) to $B(\ell_2)$. For instance this applies (rather surprisingly) to the von Neumann algebra of any free group. Nevertheless, since $B(H)$ fails the approximation property (due to Szankowski) there are $M$'s (namely $B(H)^{**}$ and certain finite examples defined using ultraproducts) that are not seemingly injective. Moreover, for $M$ to be seemingly injective it suffices to have the above factorization of $Id_M$ through $B(H)$ with $u,v$ positive (and $u$ still normal).

Key words: von Neumann algebra, injectivity, positive approximation property

中图分类号:

46L10

Gilles PISIER. SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS[J]. 数学物理学报(英文版), 2021, 41(6): 2055-2085.

Gilles PISIER. SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS[J]. Acta mathematica scientia,Series B, 2021, 41(6): 2055-2085.

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