TY - JOUR
T1 - Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology
AU - Ando, Hiroshi
AU - Haagerup, Uffe
AU - Winsløw, Carl
PY - 2016/6
Y1 - 2016/6
N2 - Based on the analysis on the Ocneanu/Groh-Raynaud ultraproducts and the Effros-Maréchal 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 ∈ vN(H), the following conditions are equivalent: (1) M has the Kirchberg's quotient weak expectation property (QWEP). (2) M is in the closure of the set inj of injective factors on H with respect to the Effros-Maréchal topology. (3) M admits an embedding i into the Ocneanu ultrapower Rω∞ of the injective III1 factor R∞ with a normal faithful conditional expectation ϵ: Rω∞ → i(M). (4) For every ϵ > 0, n ∈ ℕ and ξ1;⋯; ξn ∈ ℘M, there are k ∈ ℕ and a1;⋯; an ∈ Mk(ℂ)+ such that |〈ξi, ξj〉 - trk(aiaj)| < ϵ (1 ≤ i; j ≤ n) holds, where trk is the tracial state on Mk(ℂ), and PM is the natural cone in the standard form of M.
AB - Based on the analysis on the Ocneanu/Groh-Raynaud ultraproducts and the Effros-Maréchal 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 ∈ vN(H), the following conditions are equivalent: (1) M has the Kirchberg's quotient weak expectation property (QWEP). (2) M is in the closure of the set inj of injective factors on H with respect to the Effros-Maréchal topology. (3) M admits an embedding i into the Ocneanu ultrapower Rω∞ of the injective III1 factor R∞ with a normal faithful conditional expectation ϵ: Rω∞ → i(M). (4) For every ϵ > 0, n ∈ ℕ and ξ1;⋯; ξn ∈ ℘M, there are k ∈ ℕ and a1;⋯; an ∈ Mk(ℂ)+ such that |〈ξi, ξj〉 - trk(aiaj)| < ϵ (1 ≤ i; j ≤ n) holds, where trk is the tracial state on Mk(ℂ), and PM is the natural cone in the standard form of M.
U2 - 10.1515/crelle-2014-0005
DO - 10.1515/crelle-2014-0005
M3 - Journal article
SN - 0075-4102
VL - 715
SP - 231
EP - 250
JO - Journal fuer die Reine und Angewandte Mathematik
JF - Journal fuer die Reine und Angewandte Mathematik
ER -