Ultraproducts of von Neumann algebras

By Hiroshi Ando, Uffe Haagerup. 2012

129 citations recorded by the source corpus.

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 $M_{\omega}$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/620e8a2fe15d779ed668b3f2c7e0d26d586ebfbb

Citations in the corpus