An analogue of Connes-Haagerup approach for classification of subfactors of type III$_{\bf 1}$

Abstract

Popa proved that strongly amenable subfactors of type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$ with the same type $\mathrm{I}\mathrm{I}$ and type $\mathrm{I}\mathrm{I}\mathrm{I}$ principal graphs are completely classified by their standard invariants. In this paper, we present a different proof of this classification theorem based on Connes and Haagerup's arguments on the uniqueness of the injective factor of type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$.