Ergodic group actions with nonunique invariant means
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Abstract:
Let $M(X,G)$ be the set of $G$-invariant means on ${L^\infty }(X,\mathcal {B},P)$, where $G$ is a countable group acting ergodically as measure preserving transformations on a nonatomic probability space $(X,\mathcal {B},P)$. We show that if there exists $\mu \in M(X,G),\mu \ne P$, then $M(X,G)$ contains an isometric copy of $\beta N\backslash N$, where $\beta N\backslash N$ is considered as a subset of ${({l^\infty })^*}$. This provides an answer to a question raised by J. Rosenblatt in 1981.References
- Ching Chou, On topologically invariant means on a locally compact group, Trans. Amer. Math. Soc. 151 (1970), 443–456. MR 269780, DOI 10.1090/S0002-9947-1970-0269780-8
- Ching Chou, Topological invariant means on the von Neumann algebra $\textrm {VN}(G)$, Trans. Amer. Math. Soc. 273 (1982), no. 1, 207–229. MR 664039, DOI 10.1090/S0002-9947-1982-0664039-7
- Andrés del Junco and Joseph Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), no. 3, 185–197. MR 553340, DOI 10.1007/BF01673506
- V. G. Drinfel′d, Finitely-additive measures on $S^{2}$ and $S^{3}$, invariant with respect to rotations, Funktsional. Anal. i Prilozhen. 18 (1984), no. 3, 77 (Russian). MR 757256
- Edmond E. Granirer, Geometric and topological properties of certain $w^\ast$ compact convex subsets of double duals of Banach spaces, which arise from the study of invariant means, Illinois J. Math. 30 (1986), no. 1, 148–174. MR 822389
- V. Losert and H. Rindler, Almost invariant sets, Bull. London Math. Soc. 13 (1981), no. 2, 145–148. MR 608100, DOI 10.1112/blms/13.2.145
- G. A. Margulis, Some remarks on invariant means, Monatsh. Math. 90 (1980), no. 3, 233–235. MR 596890, DOI 10.1007/BF01295368
- G. A. Margulis, Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 383–396 (1983). MR 721730, DOI 10.1017/S014338570000167X
- Joseph Rosenblatt, Uniqueness of invariant means for measure-preserving transformations, Trans. Amer. Math. Soc. 265 (1981), no. 2, 623–636. MR 610970, DOI 10.1090/S0002-9947-1981-0610970-7
- Klaus Schmidt, Asymptotically invariant sequences and an action of $\textrm {SL}(2,\,\textbf {Z})$ on the $2$-sphere, Israel J. Math. 37 (1980), no. 3, 193–208. MR 599454, DOI 10.1007/BF02760961
- Klaus Schmidt, Amenability, Kazhdan’s property $T$, strong ergodicity and invariant means for ergodic group-actions, Ergodic Theory Dynam. Systems 1 (1981), no. 2, 223–236. MR 661821, DOI 10.1017/s014338570000924x
- Dennis Sullivan, For $n>3$ there is only one finitely additive rotationally invariant measure on the $n$-sphere defined on all Lebesgue measurable subsets, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 121–123. MR 590825, DOI 10.1090/S0273-0979-1981-14880-1
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 647-650
- MSC: Primary 43A07; Secondary 28D15
- DOI: https://doi.org/10.1090/S0002-9939-1987-0894431-7
- MathSciNet review: 894431