Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology

Abstract

Abstract Based on the analysis on the Ocneanu/Groh–Raynaud ultraproducts and the Effros–Maréchal topology on the space vN ( H ) ${\mathrm {vN}(H)}$ of von Neumann algebras acting on a separable Hilbert space H, we show that for a von Neumann algebra M ∈ vN ( H ) ${M\in \mathrm {vN}(H)}$ , 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 ℱ inj ${\mathcal {F}_{\rm inj}}$ of injective factors on H with respect to the Effros–Maréchal topology. (3) M admits an embedding i into the Ocneanu ultrapower R ∞ ω ${R_{\infty }^{\omega }}$ of the injective III 1 ${\mathrm {III}_1}$ factor R ∞ with a normal faithful conditional expectation ε : R ∞ ω → i ( M ) ${\varepsilon \colon R_{\infty }^{\omega }\rightarrow i(M)}$ . (4) For every ε > 0 ${\varepsilon >0}$ , n ∈ ℕ and ξ 1 , ... , ξ n ∈ 𝒫 M ♮ ${\xi _1,\ldots ,\xi _n\in \mathcal {P}_M^{\natural }}$ , there are k ∈ ℕ and a 1 , ... , a n ∈ M k ( ℂ ) + ${a_1,\ldots ,a_n\in M_k(\mathbb {C})_+}$ such that | 〈 ξ i , ξ j 〉 - tr k ( a i a j ) | < ε ${|\langle \xi _i,\xi _j\rangle -\operatorname{tr}_k(a_ia_j)|<\varepsilon }$ ( 1 ≤ i , j ≤ n ${1\le i,j\le n}$ ) holds, where tr k ${\operatorname{tr}_k}$ is the tracial state on M k ( ℂ ) ${M_k(\mathbb {C})}$ , and 𝒫 M ♮ ${\mathcal {P}_M^{\natural }}$ is the natural cone in the standard form of M.