Some Problems On The Hochschild Cohomology For Operator Algebras

This thesis is composed of three chapters.In chapter 1, we prove that every local 3-cocycle of a von Neumann algebra R. into dual R-bimodule S is a 3-cocyle.This partly answers the question on the local higher cocycles given by Kadison.In chapter 2, we study a kind of operator algebra acted on the Hilbert space H which is called the closed weakly reducible maximal triangular algebra, denoted by S. And we prove that the all n-th completely bounded cohomology group H_cbⁿ(S, B{H)) of S with cofficients in B(H) are trivial.In chapter 3, we introduce the notion of local G-derivation and show that if group G acts ergodically on a finite factor A, then A has nontrivial G-derivation and nontrivial local G-derivation.