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

We classify flows on AFD factors with faithful Connes-Takesaki modules. This is a generalization of classification of trace-scaling flows on the AFD ${\mathrm{I}\mathrm{I}}_{\mathrm{\infty }}$ factor, which is equivalent to the uniqueness of the AFD ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$ 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.