Ultraproducts, QWEP von Neumann Algebras, and the Effros-Mar\'echal Topology

TL;DR

This paper explores the connections between ultraproducts, the QWEP property, and the Effros-Maréchal topology on von Neumann algebras, establishing equivalences and embedding characterizations.

Contribution

It provides new equivalences for the QWEP property in von Neumann algebras using ultraproducts and the Effros-Maréchal topology, and characterizes QWEP via embeddings and approximation conditions.

Findings

QWEP is equivalent to being in the Effros-Maréchal closure of injective factors.

QWEP algebras embed into the Ocneanu ultrapower with a conditional expectation.

QWEP can be characterized by approximation of natural cones in finite-dimensional matrix algebras.

Abstract

Based on the analysis on the Ocneanu/Groh-Raynaud ultraproducts and the Effros-Mar\'echal topology on the space vN(H) of von Neumann algebras acting on a separable Hilbert space H, we show that for a von Neumann algebra M in vN(H), the following conditions are equivalent: (1) M has the Kirhcberg's quotient weak expectation property (QWEP). (2) M is in the closure of the set F_{inj of injective factors on H with respect to the Effros-Mar\'echal topology. (3) M admits an embedding i into the Ocneanu ultrapower R_{infty}^{omega} of the injective III_1 factor R_{\infty} with a normal faithful conditional expectation epsilon: R_{infty}^{omega} to i(M). (4) For every epsilon>0, natural number n, and xi_1,...,xi_n in P_M^{natural}, there is a natural number k and a_1,...,a_nin M_k(C)_+, such that |<xi_i,xi_j>-tr_k(a_ia_j)|<epsilon (1<=i,j<=n) holds, where tr_k is the tracial state on M_k(C), and P_M^{natural} is the natural cone in the standard form of M.