Commuting Squares and the Classification of Finite Depth Inclusions of AFD Type ${\mathrm{III}}_{\lambda }$ Factors, $\lambda \in (0,1)$

Abstract

We give a new proof of the classification result due to Sorm Popa that a finite depth inclusion of AFD type ${\mathrm{III}}_{\lambda }$ factors $N\subset M$, $\lambda \in (0,1)$, with a common discrete decomposition $\{N^{\infty }\subset M^{\infty },\theta \}$ is classified, up to isomorphism, by the type $\mathrm{II}$ core $N^{\infty }\subset M^{\infty }$ and the standard invariant of $\theta$.