On Mixing and Completely Mixing Properties of Positive $L^1$-Contractions of Finite Von Neumann Algebras

Abstract

Akcoglu and Suchaston proved the following result: Let $T:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$ be a positive contraction. Assume that for $z\in L^1(X,{\cf},\m)$ the sequence $(T^nz)$ converges weakly in $L^1(X,{\cf},\m)$, then either $\lim\limits_{n\to\infty}\|T^nz\|=0$ or there exists a positive function $h\in L^1(X,{\cf},\m)$, $h\neq 0$ such that $Th=h$. In the paper we prove an extension of this result in finite von Neumann algebra setting, and as a consequence we obtain that if a positive contraction of a noncommutative $L^1$-space has no non zero positive invariant element, then its mixing property implies completely mixing property one.