On the non-commutative Neveu decomposition and stochastic ergodic theorems

In this article, we prove Neveu decomposition for the action of the locally compact amenable semigroup of positive contractions on semifinite von Neumann algebras and thus, it entirely resolves the problem for the actions of arbitrary amenable semigroup on semifinite von Neumann algebras. We also prove it for amenable group actions by Markov automorphisms on any $\sigma$-finite von Neumann algebras. As an application, we obtain stochastic ergodic theorem for actions of $\mathbb{Z}_+^d$ and $\mathbb{R}_+^d$ for $d \in \mathbb{N}$ by positive contractions on $L^1$-spaces associated with a finite von Neumann algebra. It yields the first ergodic theorem for positive contraction on non-commutative $L^1$-spaces beyond the Danford-Schwartz category.