On mixing and completely mixing properties of positive $L^1$-contractions of finite von Neumann algebras

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