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

Akcoglu and Suchaston proved the following result: Let $T:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$beapositivecontraction\mathrm{.}Assumethatfor$z\in L^1(X,{\cf},\m)$thesequence$(T^nz)$convergesweaklyin$L^1(X,{\cf},\m)$, then either $\underset{n\to \mathrm{\infty }}{\lim }\mathrm{\parallel }{T}^{n}z\mathrm{\parallel }=0$ or there exists a positive function $h\in L^1(X,\text{cf},\text{m})$, $h\mathrm{\ne }0$ 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.