Ultraproducts of von Neumann algebras

TL;DR

This paper unifies different notions of ultraproducts of von Neumann algebras, revealing new phenomena in type III factors and connecting ultraproduct actions with modular automorphisms.

Contribution

It shows that Ocneanu ultraproducts are corners of Groh-Raynaud ultraproducts and establishes their properties, especially for type III factors, addressing recent open problems.

Findings

Ocneanu ultraproduct is a corner of Groh-Raynaud ultraproduct.

Ultrapower of a Type III₀ factor is never a factor.

Confirmed the connection between the relative commutant and Connes' asymptotic centralizer.

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 $M_{\omega}$.