Markov operators and quasi-Stonian spaces

Atalla, Robert E.

doi:10.1090/S0002-9939-1981-0614888-0

Markov operators and quasi-Stonian spaces
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Proc. Amer. Math. Soc. 82 (1981), 613-618 Request permission

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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