A Classification of Flows on AFD Factors with Faithful Connes--Takesaki Modules

TL;DR

This paper classifies flows on approximately finite-dimensional (AFD) factors with faithful Connes-Takesaki modules, extending previous classifications and establishing the Rohlin property for such flows, thus advancing the understanding of automorphism groups on type III factors.

Contribution

It generalizes the classification of trace-scaling flows on AFD II_ factors to flows with faithful Connes-Takesaki modules on AFD factors, showing they possess the Rohlin property.

Findings

Flows with faithful Connes-Takesaki modules have the Rohlin property.

A classification of flows on AFD factors with faithful modules is achieved.

The results extend Izumi's work on compact group actions to  flows.

Abstract

We classify flows on AFD factors with faithful Connes-Takesaki modules. This is a generalization of classification of trace-scaling flows on the AFD $\mathrm{II}_\infty$ factor, which is equivalent to the uniqueness of the AFD $\mathrm{III}_1$ factor. In order to do this, we show that a flow on an AFD factor with faithful Connes-Takesaki module has the Rohlin property, which gives a partial answer to a characterization problem of the Rohlin property posed by Masuda-Tomatsu. It is also possible to think of this result as an $\mathbf{R}$-version of Izumi's result about compact group actions on type III factors with faithful Connes-Takesaki modules.