1 Department of Mathematical Sciences, Faculty of Science, Københavns Universitet2 Department of Mathematical Sciences, Faculty of Science, Københavns Universitet
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 MωMω introduced by Ocneanu is a corner of the ultraproduct ∏ωM∏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 (<img height="20" border="0" style="vertical-align:bottom" width="90" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022123614001335-si3.gif">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 MωMω 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 MωMω and Connes' asymptotic centralizer algebra MωMω.
Journal of Functional Analysis, 2014, Vol 266, Issue 12, p. 6842-6913