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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
##### Keywords:
ultraproducts; type $$\mathrm {III}$$ factors
##### Citations:
Zbl 0608.46035; Zbl 0551.46046; Zbl 1029.46102; Zbl 1252.46059
Full Text:
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