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 $\alpha$ of a group $G$ on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$, $0<\lambda<1$, we study first the cohomology group of a group with the unitary group of an abelian von Neumann algebra as a coefficient group and establish a technique to reduce the coefficient group to the torus $\mathsf T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a countable discrete amenable group on an AFD factor of type $\mathrm{III}_\lambda$, sharpening the result in [17, §4]. The periodicity of the flow of weights on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$ allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\mathrm{Aut}(\mathcal M)_{m}$ relative to the modulus homomorphism $m=\mod\colon \mathrm{Aut}(\mathcal M)\to \mathsf R/T'\mathsf Z$. We then discuss the reduced modified HJR-exact sequence, which allows us to describe the invariant of outer action $\alpha$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $H_{m,s}^{out}(G,N,\mathsf T)$ 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 $\mathcal M$ of type $\mathrm{III}_{\lambda}$