Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (6): 2055-2085.doi: 10.1007/s10473-021-0616-0

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Gilles PISIER   

  1. Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
  • Received:2021-04-06 Revised:2021-09-24 Online:2021-12-25 Published:2021-12-27

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