On the non-commutative Neveu decomposition and ergodic theorems for amenable group action

Abstract

We prove non-commutative analogue of Neveu decomposition for actions of locally compact amenable groups on finite von Neumann algebras. In addition, if we assume G=Z+ or G is a locally compact group of polynomial growth with a symmetric compact generating set V, then for a state preserving action α of G on a finite von Neumann algebra M, we show that the ergodic averages associated with the predual action on M⁎ corresponding to the Følner sequence {Kn}n∈N (where Kn={0,1,…n−1} for G=Z+ and Kn=Vn otherwise) converges bilateral almost uniformly. A maximal ergodic inequality is also proved in the process. At the end, using these results, we establish the stochastic ergodic theorem.