On Cocycle Actions of Non-commutative Bernoulli Shifts

In this paper we investigate the cocycle actions of non-commutative Bernoulli shifts for a countable discrete group G on the AFD II1-factor N = ⊗g∈GMn(C) or ⊗g∈GR, where R is the AFD II1-factor. We show that if G contains some non-amenable exact group, then the fixed point algebra of any its cocycle action is always atomic. We also give another proof of Popa’s cocycle vanishing theorem [15] in this special case; We will show that if G has relative property T and contains some non-amenable exact group, then all unitary cocycles are cohomologous to characters. In our proof, Ozawa’s theorem[11] and Popa’s argument[16] play a crucial role.