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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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

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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Additional 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