Noncommutative topological dynamics. I

Avitzour, Daniel

doi:10.1090/S0002-9947-1984-99926-0

Noncommutative topological dynamics. I

Author: Daniel Avitzour
Journal: Trans. Amer. Math. Soc. 282 (1984), 109-119
MSC: Primary 46L55; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9947-1984-99926-0
Part II: Trans. Amer. Math. Soc. (1) (1984), 121-135
MathSciNet review: 728705
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Abstract: Ergodicity and minimality are defined for $C^{\ast}$ -flows where is a group acting on a $C^{\ast}$ -algebra by *-automorphisms. Elementary properties are proved and several examples are given. In particular, an example shows that there are arbitrarily large $C^{\ast}$ -algebras admitting a minimal action of the integers.

[1] E. M. Alfsen and F. W. Schultz, Noncommutative spectral theory for affine function spaces on convex sets, Mem. Amer. Math. Soc. No. 172 (1976).
[2] Daniel Avitzour, Free products of 𝐶*-algebras, Trans. Amer. Math. Soc. 271 (1982), no. 2, 423–435. MR 654842, https://doi.org/10.1090/S0002-9947-1982-0654842-1
[3] -, Noncommutative topological dynamics. II, Trans. Amer. Math. Soc. 282(1984), 121-135.
[4] Jacques Dixmier, Les 𝐶*-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
[5] Edward G. Effros, Order ideals in a 𝐶*-algebra and its dual, Duke Math. J. 30 (1963), 391–411. MR 0151864
[6] H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477–515. MR 0157368, https://doi.org/10.2307/2373137
[7] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 0213508, https://doi.org/10.1007/BF01692494
[8] Alain Guichardet, Systèmes dynamiques non commutatifs, Société Mathématique de France, Paris, 1974 (French). With an English summary; Séminaire rédigé par A. Guichardet d’après des exposés faits au Séminaire Dang Ngoc—Guichardet 1972-1973 par C. Cahmbon, F. Combes, N. Dang Ngoc, M. Enock, A. Guichardet, F. Ledrappier, J. C. Marcuard, O. Maréchal, M. Samuélidès, J. L. Sauvageot, J. M. Schwartz, J. P. Thouvenot; Astérisque, Nos. 13-14. MR 0352997
[9] Harvey B. Keynes and James B. Robertson, On ergodicity and mixing in topological transformation groups, Duke Math. J. 35 (1968), 809–819. MR 0234441
[10] Harvey B. Keynes and James B. Robertson, Eigenvalue theorems in topological transformation groups, Trans. Amer. Math. Soc. 139 (1969), 359–369. MR 0237748, https://doi.org/10.1090/S0002-9947-1969-0237748-5
[11] Diane Laison and Gary Laison, Topological dynamics on 𝐶*-algebras, Trans. Amer. Math. Soc. 204 (1975), 197–205. MR 0358365, https://doi.org/10.1090/S0002-9947-1975-0358365-5
[12] Gert K. Pedersen, 𝐶*-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
[13] William A. Veech, Finite group extensions of irrational rotations, Israel J. Math. 21 (1975), no. 2-3, 240–259. Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 1974). MR 0396913, https://doi.org/10.1007/BF02760801