On free actions, minimal flows, and a problem by Ellis
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- by Vladimir G. Pestov PDF
- Trans. Amer. Math. Soc. 350 (1998), 4149-4165 Request permission
Abstract:
We exhibit natural classes of Polish topological groups $G$ such that every continuous action of $G$ on a compact space has a fixed point, and observe that every group with this property provides a solution (in the negative) to a 1969 problem by Robert Ellis, as the Ellis semigroup $E(U)$ of the universal minimal $G$-flow $U$, being trivial, is not isomorphic with the greatest $G$-ambit. Further refining our construction, we obtain a Polish topological group $G$ acting freely on the universal minimal flow $U$ yet such that $\mathcal {S} (G)$ and $E(U)$ are not isomorphic. We also display Polish topological groups acting effectively but not freely on their universal minimal flows. In fact, we can produce examples of groups of all three types having any prescribed infinite weight. Our examples lead to dynamical conclusions for some groups of importance in analysis. For instance, both the full group of permutations $S(X)$ of an infinite set, equipped with the pointwise topology, and the unitary group $U(\mathcal {H})$ of an infinite-dimensional Hilbert space with the strong operator topology admit no free action on a compact space, and the circle $\mathbb {S}^{1}$ forms the universal minimal flow for the topological group $\mathrm {Homeo} _{+}(\mathbb {S}^{1})$ of orientation-preserving homeomorphisms. It also follows that a closed subgroup of an amenable topological group need not be amenable.References
- A. V. Arhangel′skiĭ, On the relations between invariants of topological groups and their subspaces, Uspekhi Mat. Nauk 35 (1980), no. 3(213), 3–22 (Russian). International Topology Conference (Moscow State Univ., Moscow, 1979). MR 580615
- Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR 956049
- Paul Bankston, Ultraproducts in topology, General Topology and Appl. 7 (1977), no. 3, 283–308. MR 458351
- Robert B. Brook, A construction of the greatest ambit, Math. Systems Theory 4 (1970), 243–248. MR 267038, DOI 10.1007/BF01691107
- W. W. Comfort and Kenneth A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483–496. MR 207886
- Dikran N. Dikranjan, Ivan R. Prodanov, and Luchezar N. Stoyanov, Topological groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 130, Marcel Dekker, Inc., New York, 1990. Characters, dualities and minimal group topologies. MR 1015288
- Robert Ellis, Universal minimal sets, Proc. Amer. Math. Soc. 11 (1960), 540–543. MR 117716, DOI 10.1090/S0002-9939-1960-0117716-1
- Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
- Edward D. Gaughan, Topological group structures of infinite symmetric groups, Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 907–910. MR 215940, DOI 10.1073/pnas.58.3.907
- B. R. Gelbaum, Free topological groups, Proc. Amer. Math. Soc. 12 (1961), 737–743. MR 140607, DOI 10.1090/S0002-9939-1961-0140607-8
- Shmuel Glasner, Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976. MR 0474243
- —, Structure theory as a tool in topological dynamics, Lectures given during the Descriptive Set Theory and Ergodic Theory Joint Workshop (Luminy, June 1996), Tel Aviv University preprint, 26 pp.
- —, On minimal actions of Polish groups, Tel Aviv University preprint, October 1996, 6 pp.
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey theory, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1980. MR 591457
- E. Granirer, Extremely amenable semigroups. II, Math. Scand. 20 (1967), 93–113. MR 212551, DOI 10.7146/math.scand.a-10825
- Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- Wojchiech Herer and Jens Peter Reus Christensen, On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann. 213 (1975), 203–210. MR 412369, DOI 10.1007/BF01350870
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- E. Makai Jr., Notes on real closed fields, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13 (1970), 35–55 (1971). MR 309910
- M. G. Megrelishvili, Compactification and factorization in the category of $G$-spaces, Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988) World Sci. Publ., Teaneck, NJ, 1989, pp. 220–237. MR 1047903
- M. G. Megrelishvili, Free topological $G$-groups, New Zealand J. Math. 25 (1996), no. 1, 59–72. MR 1398366
- Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630–641. MR 270356
- Sidney A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145–160. MR 259010, DOI 10.1017/S0004972700041393
- Sidney A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145–160. MR 259010, DOI 10.1017/S0004972700041393
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- V. G. Pestov, Embeddings and condensations of topological groups, Mat. Zametki 31 (1982), no. 3, 443–446, 475 (Russian). MR 652848
- —, Topological groups and algebraic envelopes of topological spaces, Ph.D. thesis, Moscow State University, Moscow, 1983, 78 pp. (in Russian).
- V. G. Pestov, A criterion for the balance of a locally compact group, Ukrain. Mat. Zh. 40 (1988), no. 1, 127–129, 136 (Russian); English transl., Ukrainian Math. J. 40 (1988), no. 1, 109–111. MR 936417, DOI 10.1007/BF01056462
- —, Epimorphisms of topological groups by way of topological dynamics, New Zealand J. Math. 26 (1997), 257–262.
- D. B. Shakhmatov, Character and pseudocharacter in minimal topological groups, Mat. Zametki 38 (1985), no. 6, 908–914, 959 (Russian). MR 823429
- Silviu Teleman, Sur la représentation linéaire des groupes topologiques, Ann. Sci. Ecole Norm. Sup. (3) 74 (1957), 319–339 (French). MR 0097458
- V. V. Uspenskiĭ, A universal topological group with a countable basis, Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 86–87 (Russian). MR 847156
- V. V. Uspenskij, The epimorphism problem for Hausdorff topological groups, Topology Appl. 57 (1994), no. 2-3, 287–294. MR 1278029, DOI 10.1016/0166-8641(94)90055-8
- William A. Veech, Topological dynamics, Bull. Amer. Math. Soc. 83 (1977), no. 5, 775–830. MR 467705, DOI 10.1090/S0002-9904-1977-14319-X
- Jan de Vries, On the existence of $G$-compactifications, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 3, 275–280 (English, with Russian summary). MR 644661
- J. de Vries, Elements of topological dynamics, Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1249063, DOI 10.1007/978-94-015-8171-4
Additional Information
- Vladimir G. Pestov
- Affiliation: School of Mathematical and Computing Sciences, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand
- MR Author ID: 138420
- Email: pestovv@member.ams.org
- Received by editor(s): August 19, 1996
- Additional Notes: Research partially supported by the New Zealand Ministry of Research, Science and Technology through the project “Dynamics in Function Spaces” of the International Science Linkages Fund.
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 4149-4165
- MSC (1991): Primary 54H20
- DOI: https://doi.org/10.1090/S0002-9947-98-02329-0
- MathSciNet review: 1608494
Dedicated: Dedicated to my teacher, Professor Alexander V. Arhangel’skiĭ, on his 60th birthday