SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS

doi:10.1007/s10473-021-0616-0

Abstract

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

CLC Number:

46L10

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

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