Outer Actions of a Discrete Amenable Group on Approximately Finite Dimensional Factors II, the III$_\lambda$-Case, $\lambda\neq 0$

Abstract

To study outer actions $\a$ of a group $G$ on a factor $\sM$ of type {\threel}, $0<\la<1$, we study first the cohomology group of a group with the unitary group of an abelian {\vna} as a coefficient group and establish a technique to reduce the coefficient group to the torus $\T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a {\cdag} on an AFD factor of type {\threel}, sharpening the result in \cite{KtT2: \S4}. The periodicity of the flow of weights on a factor $\sM$ of type {\threel} allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\Aut(\sM)_\txm$ relative to the modulus homomorphism $\txm=\mod: \Aut(\sM)\mapsto \rt'z$. We then discuss the reduced {\mhjr}, which allows us to describe the invariant of outer action $\a$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $\tH_{\txm, \fs}^\out(G, N, \T)$ of modular obstructions is a compact abelian group. Making use of these reductions, we prove the classification result of outer actions of $G$ on an {\AFD} factor $\sM$ of type \threel.