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

  • Hiroshi Ando EMAIL logo , Uffe Haagerup und Carl Winsløw
Veröffentlicht/Copyright: 7. März 2014
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Journal für die reine und angewandte Mathematik (Crelles Journal)
Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2016 Heft 715

Abstract

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 ℱ 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 of the injective III 1 factor R∞ with a normal faithful conditional expectation ε:R∞ω→i(M). (4) For every ε>0, n ∈ ℕ and ξ1,...,ξn∈𝒫M♮, 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.

Funding source: ERC

Award Identifier / Grant number: OAFPG 27731

Funding statement:

The authors thank Valerio Capraro, Masaki Izumi and Narutaka Ozawa for useful comments on the paper and information about literature.

Received: 2013-11-13
Revised: 2013-12-17
Published Online: 2014-3-7
Published in Print: 2016-6-1

© 2016 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity
  3. Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras
  4. Mixing operators and small subsets of the circle
  5. Singularities on the base of a Fano type fibration
  6. Stable representation homology and Koszul duality
  7. An inverse spectral problem for a star graph of Krein strings
  8. Uniqueness of self-similar shrinkers with asymptotically cylindrical ends
  9. Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology
https://doi.org/10.1515/crelle-2014-0005

Artikel in diesem Heft

  1. Frontmatter
  2. Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity
  3. Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras
  4. Mixing operators and small subsets of the circle
  5. Singularities on the base of a Fano type fibration
  6. Stable representation homology and Koszul duality
  7. An inverse spectral problem for a star graph of Krein strings
  8. Uniqueness of self-similar shrinkers with asymptotically cylindrical ends
  9. Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology
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