Markov operators and quasi-Stonian spaces
HTML articles powered by AMS MathViewer
- by Robert E. Atalla PDF
- 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.References
- T. AndΓ΄, Invariante Masse positiver Kontraktionen in $C(X)$, Studia Math. 31 (1968), 173β187 (German). MR 239046, DOI 10.4064/sm-31-2-173-187
- Robert E. Atalla, Measure theoretic behavior of closed sets, Comment. Math. Univ. Carolin. 19 (1978), no.Β 4, 697β703. MR 518181
- Robert E. Atalla, Generalized almost-convergence vs. matrix summability, Colloq. Math. 47 (1982), no.Β 1, 103β111. MR 679391, DOI 10.4064/cm-47-1-103-111 M. Day, Normed linear spaces, 1st ed., Springer, Berlin and New York, 1962.
- Hugh Gordon, The maximal ideal space of a ring of measurable functions, Amer. J. Math. 88 (1966), 827β843. MR 201961, DOI 10.2307/2373081
- Stuart P. Lloyd, On the mean ergodic theorem of Sine, Proc. Amer. Math. Soc. 56 (1976), 121β126. MR 451007, DOI 10.1090/S0002-9939-1976-0451007-6
- Stuart P. Lloyd, Two lifting theorems, Proc. Amer. Math. Soc. 42 (1974), 128β134. MR 328588, DOI 10.1090/S0002-9939-1974-0328588-4
- Ry\B{o}tar\B{o} Sat\B{o}, Invariant measures and ergodic theorems for positive operators on $C(X)$ with $X$ quasi-Stonian, Math. J. Okayama Univ. 22 (1980), no.Β 1, 77β90. MR 573676
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
- G. L. Seever, Measures on $F$-spaces, Trans. Amer. Math. Soc. 133 (1968), 267β280. MR 226386, DOI 10.1090/S0002-9947-1968-0226386-5
- Robert Sine, Geometric theory of a single Markov operator, Pacific J. Math. 27 (1968), 155β166. MR 240281
- Robert Sine, A mean ergodic theorem, Proc. Amer. Math. Soc. 24 (1970), 438β439. MR 252605, DOI 10.1090/S0002-9939-1970-0252605-X
- A. I. Veksler, $P$-sets in topological spaces, Dokl. Akad. Nauk SSSR 193 (1970), 510β513 (Russian). MR 0279759
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 613-618
- MSC: Primary 47A35; Secondary 54G05
- DOI: https://doi.org/10.1090/S0002-9939-1981-0614888-0
- MathSciNet review: 614888