G-kernel of amenable discrete groups on AFD factors of type III

摘要。对于每一个因子M，我们将一个称为内模结构的不变量Ob_m（M）关联为存在于“第三”上同调群中的上同调不变量：H^<out>_&lt;α，S&gt;（Out（M）× R，H^1_θ（R，U（C），U（C））其中{C，R，θ}是M上的权流。设α是可数离散群G在M上的一个外作用，则它的模mod（α）∈ Hom（G，Aut_θ（C）），N = α^<-1>（Cnt_r（M））和称为α的模阻塞的拉回Ob_m（α）= α^*（Ob_m（M））∈ H^<out>_&lt;α，S&gt;（G×R，N，U（C））是外作用α的外共轭类的不变量.我们证明了：若因子M是近似有限维的且G是顺从的，则这些不变量唯一地确定α的外共轭类，且每个不变量都作为G在M上的一个外作用α a的内变量出现.

ABSTRACT. To each factor M, we associate an invariant Ob_m(M) to be called the intrinsic modular abstruction as a cohomological invariant which lives in the "third" cohomology group :H^<out>_<α,S>(Out(M)×R,H^1_θ(R,U(C),U(C))where {C,R,θ} is the flow of weights on M. If α is an outer action of a countable discrete group G on M, then its modulus mod(α) ∈ Hom(G, Aut_θ(C)), N = α^<-1>(Cnt_r(M)) and the pull backOb_m(α) = α^*(Ob_m(M)) ∈ H^<out>_<α,S>(G×R,N,U(C))to be called the modular obstruction of α are invariants of the outer conjugacy class of the outer action α.We prove that if the factor M is approximately finite dimensional and G is amenable, then the invariants uniquely determine the outer conjugacy class of α and the every invariant occurs as the in variant of an outer action α a of G on M.