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

Authors: Farruh Mukhamedov, Seyit Temir and Hasan Akin
Journal: Proc. Amer. Math. Soc. 134 (2006), 843-850
MSC (2000): Primary 47A35, 28D05
Published electronically: July 20, 2005
MathSciNet review: 2180902
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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 $\lim\limits_{n\to\infty}\Vert T^nz\Vert=0$ 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.

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Additional Information

Farruh Mukhamedov
Affiliation: Department of Mechanics and Mathematics, National University of Uzbekistan, Vuzgorodok, 700095, Tashkent, Uzbekistan

Seyit Temir
Affiliation: Department of Mathematics, Arts and Science Faculty, Harran University, 63200, Şanliurfa, Turkey

Hasan Akin
Affiliation: Department of Mathematics, Arts and Science Faculty, Harran University, 63200, Şanliurfa, Turkey

Keywords: Positive contraction, mixing, completely mixing, von Neumann algebra
Received by editor(s): June 30, 2004
Received by editor(s) in revised form: October 21, 2004
Published electronically: July 20, 2005
Additional Notes: This work was supported by NATO-TUBITAK PC-B programme
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society

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