Resumen de Outer actions of a discrete amenable group on approximately finite dimensional factors II, the III [l] -case, [L] # 0
Yoshikazu Katayama, Masamichi Takesaki
To study outer actions a of a group G on a factor M of type III[l], 0 < [L] < 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[l], sharpening the result in [17, §4]. The periodicity of the flow of weights on a factorMof type III[l] 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 = mod:Aut(M) _ R/T _Z.We then discuss the reduced modified HJR-exact sequence, 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 Hout m,_(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 factorM of type III[l].
