Ultraproducts of von Neumann algebras

Abstract

We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct Mω introduced by Ocneanu is a corner of the ultraproduct ∏ωM introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state (σtφω=(σtφ)ω). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower Mω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in Mω and Connes' asymptotic centralizer algebra Mω.