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Ultraproducts of von Neumann algebras. (English) Zbl 1305.46049
Summary: 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 \(M^\omega\) introduced by A. Ocneanu [Action of discrete amenable groups on von Neumann algebras. Berlin etc.: Springer-Verlag (1985; Zbl 0608.46035)] is a corner of the ultraproduct \(\prod^\omega M\) introduced by U. Groh [J. Oper. Theory 11, 395–404 (1984; Zbl 0551.46046)] and Y. Raynaud [J. Oper. Theory 48, No. 1, 41–68 (2002; Zbl 1029.46102)]. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state \(\varphi\) of \(M\) on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state (\(\sigma_t^{\varphi^\omega} = (\sigma_t^\varphi)^\omega\)). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type \(\mathrm {III}\) factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower \(M^\omega\) of a Type \(\mathrm {III}_{0}\) factor is never a factor. Moreover, we settle in the affirmative a recent problem by Y. Ueda [Adv. Math. 228, No. 5, 2647–2671 (2011; Zbl 1252.46059)] about the connection between the relative commutant of \(M\) in \(M^\omega\) and Connes’ asymptotic centralizer algebra \(M_\omega\).

MSC:
46L10 General theory of von Neumann algebras
46L36 Classification of factors
46M07 Ultraproducts in functional analysis
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