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2013 Weak ergodicity of nonhomogeneous Markov chains on noncommutative $L^1$-spaces
Farrukh Mukhamedov
Banach J. Math. Anal. 7(2): 53-73 (2013). DOI: 10.15352/bjma/1363784223


In this paper we study certain properties of Dobrushin's ergodicity coefficient for stochastic operators defined on noncommutative $L^1$-spaces associated with semi-finite von Neumann algebras. Such results extends the well-known classical ones to a noncommutative setting. This allows us to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. We provide a sufficient conditions for such processes to satisfy the weak ergodicity. Moreover, a necessary and sufficient condition is given for the satisfaction of the $L^1$-weak ergodicity of NDMP. It is also provided an example showing that $L^1$-weak ergodicity is weaker that weak ergodicity. We applied the main results to several concrete examples of noncommutative NDMP.


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Farrukh Mukhamedov . "Weak ergodicity of nonhomogeneous Markov chains on noncommutative $L^1$-spaces." Banach J. Math. Anal. 7 (2) 53 - 73, 2013.


Published: 2013
First available in Project Euclid: 20 March 2013

zbMATH: 1276.47012
MathSciNet: MR3039939
Digital Object Identifier: 10.15352/bjma/1363784223

Primary: 47A35
Secondary: 28D05

Keywords: $L^1$-weak ergodic , Dobrushin ergodicity cofficient , uniform ergodic , von Neumann algebra , weak ergodic

Rights: Copyright © 2013 Tusi Mathematical Research Group


Vol.7 • No. 2 • 2013
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