On mixing and completely mixing properties of positive 𝐿¹-contractions of finite von Neumann algebras

Mukhamedov, Farruh; Temir, Seyit; Akin, Hasan

doi:10.1090/S0002-9939-05-08072-X

On mixing and completely mixing properties of positive $L^1$-contractions of finite von Neumann algebras
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by Farruh Mukhamedov, Seyit Temir and Hasan Akin

Proc. Amer. Math. Soc. 134 (2006), 843-850

DOI: https://doi.org/10.1090/S0002-9939-05-08072-X

Published electronically: July 20, 2005

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Abstract:

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 $\lim \limits _{n\to \infty }\|T^nz\|=0$ 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.

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