Ultraproducts of von Neumann algebras

Abstract

We study several notions of ultraproducts of von Neumann algebras from a unifying viewpoint. In particular, we show that for a sigma-finite von Neumann algebra $M$, the ultraproduct $M^{\omega}$ introduced by Ocneanu is a corner of the ultraproduct $\prod^{\omega}M$ introduced by Groh and Raynaud. 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 phenomena of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower $M^{\omega}$ of a Type III$_0$ 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 $M^{\omega}$ and Connes' asymptotic centralizer algebra $M_{\omega}$.