Outer actions of a discrete amenable group on approximately finite dimensional factors II, the III_λ-case, λ eq 0 - Katayama, Yoshikazu; Takesaki, Masamichi (Mathematica Scandinavica, 2007)
To study outer actions α of a group G on a factor M of type III_λ, 0<λ<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 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 III_λ, sharpening the result in [17, §4]. The periodicity of the flow of weights on a factor M of type III_λ 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(M)_m relative to the modulus homomorphism m=Aut(M)→ R/T’Z. We then discuss the reduced modified HJR-exact sequence, which allows us to describe the invariant of outer action α in a simpler form than the one for a general AFD factor: for example, the cohomology group H_m,s^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 M of type III_λ.