Series of Lectures

Speaker: **Reiji Tomatsu ** (Hokkaido Univ.)

Title: Introduction to the Ando-Haagerup theory

Date/Time/Room |

November 5 (Tue.), 2013, 3:30-5:30, 118 Math. Sci. Building |

November 6 (Wed.), 2013, 10:00-12:00, 122 Math. Sci. Building |

November 7 (Thu.), 2013, 3:30-5:30, 123 Math. Sci. Building |

November 8 (Fri.), 2013, 10:00-12:00, 122 Math. Sci. Building |

Abstract:
An ultraproduct von Neumann algebra (e.g. M_{ω}, M^{ω}) plays
an important role in study of various properties of a von Neumann algebra
or classification of group actions.

The work due to H. Ando and U. Haagerup (2012) improves our understanding of ultraproduct von Neumann algebras. Among other things, their main theorem, which states that the modular automorphism group of the ultraproduct state is equal to the ultraproduct of the modular automorphism group, at last enables us to "correctly" treat ultraproducts for a type III von Neumann algebra.

In this talk, I will select some topics including the following 1, 2 and 3 from their work and present self-contained proofs.

1. Groh-Raynaud ultraproducts vs Ocneanu ultraproducts.

2. The modular automorphism group of the ultraproduct state = the ultraproduct of the modular automorphism group.

3. Ueda's question
(If M is a full factor, then M'∩M^{ω}=**C**?)

As background knowledge, the Tomita-Takesaki theory and theory of standard forms are enough.