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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Markov operators and quasi-Stonian spaces


Author: Robert E. Atalla
Journal: Proc. Amer. Math. Soc. 82 (1981), 613-618
MSC: Primary 47A35; Secondary 54G05
MathSciNet review: 614888
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Abstract: Let $ X$ be a quasi-stonian space, and let $ T$ be a $ \sigma $-additive Markov operator on $ C(X)$. Ando proved that if all $ T$-invariant probabilities are $ \sigma $-additive, then $ T$ is strongly ergodic (and the space of fixed points is finite-dimensional). We prove that if the set of $ \sigma $-additive $ T$-invariant probabilities is weak-* dense in the set of all $ T$-invariant probabilities, then $ T$ is strongly ergodic. This result is easy in case $ X$ is hyperstonian. Our method of proof is to use an idea of Gordon to "hyperstonify" part of our quasi-stonian space.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0614888-0
PII: S 0002-9939(1981)0614888-0
Article copyright: © Copyright 1981 American Mathematical Society