Seemingly injective von Neumann algebras

DOI10.1007/S10473-021-0616-0arXiv2010.13743WikidataQ114227631 ScholiaQ114227631MaRDI QIDQ6352277

Gilles Pisier

Publication date: 26 October 2020

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 $M$ ${\displaystyle }$Id_M=vu: M{�uildrel uoverlongrightarrow} B(H) {�uildrel voverlongrightarrow} M${\displaystyle }$ 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 $B(el{l}_2)$. 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 $I{d}_M$ through $B(H)$ with $u,v$ positive (and $u$ still normal).

Mathematics Subject Classification ID

General theory of von Neumann algebras (46L10) Spaces of operators; tensor products; approximation properties (46B28) Operator spaces and completely bounded maps (46L07) Convex sets and cones of operators (47L07)

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