## 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.## References

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## Bibliographic Information

**Farruh Mukhamedov**- Affiliation: Department of Mechanics and Mathematics, National University of Uzbekistan, Vuzgorodok, 700095, Tashkent, Uzbekistan
- Email: far75m@yandex.ru
**Seyit Temir**- Affiliation: Department of Mathematics, Arts and Science Faculty, Harran University, 63200, Şanliurfa, Turkey
- Email: seyittemir67@hotmail.com
**Hasan Akin**- Affiliation: Department of Mathematics, Arts and Science Faculty, Harran University, 63200, Şanliurfa, Turkey
- MR Author ID: 734288
- ORCID: 0000-0001-6447-4035
- Email: hasanakin69@hotmail.com
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 843-850 - MSC (2000): Primary 47A35, 28D05
- DOI: https://doi.org/10.1090/S0002-9939-05-08072-X
- MathSciNet review: 2180902