# zbMATH — the first resource for mathematics

On free actions, minimal flows, and a problem by Ellis. (English) Zbl 0911.54034
This is an important article. The author exhibits natural classes of Polish topological groups $$G$$ such that every continuous action of $$G$$ on a compact space has a fixed point, and observes 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. There are also many other interesting results.
Reviewer: T.S.Wu (Cleveland)

##### MSC:
 54H20 Topological dynamics (MSC2010) 54E45 Compact (locally compact) metric spaces
Full Text:
##### References:
 [1] A. V. Arhangel$$^{\prime}$$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). [2] 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. · Zbl 0654.54027 [3] Paul Bankston, Ultraproducts in topology, General Topology and Appl. 7 (1977), no. 3, 283 – 308. · Zbl 0364.54005 [4] Robert B. Brook, A construction of the greatest ambit, Math. Systems Theory 4 (1970), 243 – 248. · Zbl 0205.04301 [5] W. W. Comfort and Kenneth A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483 – 496. · Zbl 0214.28502 [6] 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. · Zbl 0687.22001 [7] Robert Ellis, Universal minimal sets, Proc. Amer. Math. Soc. 11 (1960), 540 – 543. · Zbl 0102.38002 [8] Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. · Zbl 0193.51502 [9] Edward D. Gaughan, Topological group structures of infinite symmetric groups, Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 907 – 910. · Zbl 0153.04301 [10] B. R. Gelbaum, Free topological groups, Proc. Amer. Math. Soc. 12 (1961), 737 – 743. · Zbl 0106.02604 [11] Shmuel Glasner, Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976. · Zbl 0322.54017 [12] -, 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. [13] -, On minimal actions of Polish groups, Tel Aviv University preprint, October 1996, 6 pp. [14] M. I. Graev, Free topological groups, Amer. Math. Soc. Transl. 35 (1951), 61 pp. [15] Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey theory, John Wiley & Sons, Inc., New York, 1980. Wiley-Interscience Series in Discrete Mathematics; A Wiley-Interscience Publication. · Zbl 0455.05002 [16] E. Granirer, Extremely amenable semigroups. II, Math. Scand. 20 (1967), 93 – 113. · Zbl 0204.03501 [17] 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. · Zbl 0174.19001 [18] F. Hausdorff, Grundzüge der Mengenlehre, Leipzig, 1914. (reprint) · JFM 45.0123.01 [19] 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. · Zbl 0311.28002 [20] 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. · Zbl 0416.43001 [21] J. L. Kelley, General Topology, D. van Nostrand, Inc., Princeton, NJ, 1955. · Zbl 0066.16604 [22] E. Makai Jr., Notes on real closed fields, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13 (1970), 35 – 55 (1971). · Zbl 0208.05102 [23] M. G. Megrelishvili, Compactification and factorization in the category of \?-spaces, Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988) World Sci. Publ., Teaneck, NJ, 1989, pp. 220 – 237. [24] M. G. Megrelishvili, Free topological \?-groups, New Zealand J. Math. 25 (1996), no. 1, 59 – 72. · Zbl 0848.22003 [25] Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630 – 641. · Zbl 0219.22003 [26] Sidney A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145 – 160. , https://doi.org/10.1017/S0004972700041393 Sidney A. Morris, Varieties of topological groups. II, Bull. Austral. Math. Soc. 2 (1970), 1 – 13. , https://doi.org/10.1017/S0004972700041563 Sidney A. Morris, Varieties of topological groups. III, Bull. Austral. Math. Soc. 2 (1970), 165 – 178. · Zbl 0186.32901 [27] Sidney A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145 – 160. , https://doi.org/10.1017/S0004972700041393 Sidney A. Morris, Varieties of topological groups. II, Bull. Austral. Math. Soc. 2 (1970), 1 – 13. , https://doi.org/10.1017/S0004972700041563 Sidney A. Morris, Varieties of topological groups. III, Bull. Austral. Math. Soc. 2 (1970), 165 – 178. · Zbl 0186.32901 [28] B. H. Neumann, On ordered groups, Amer. J. Math. 71 (1949), 1-18. · Zbl 0031.34201 [29] -, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202-252. · Zbl 0035.30401 [30] V. G. Pestov, Embeddings and condensations of topological groups, Mat. Zametki 31 (1982), no. 3, 443 – 446, 475 (Russian). [31] -, Topological groups and algebraic envelopes of topological spaces, Ph.D. thesis, Moscow State University, Moscow, 1983, 78 pp. (in Russian). [32] 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. · Zbl 0648.22003 [33] -, Epimorphisms of topological groups by way of topological dynamics, New Zealand J. Math. 26 (1997), 257-262. CMP 98:07 [34] D. B. Shakhmatov, Character and pseudocharacter in minimal topological groups, Mat. Zametki 38 (1985), no. 6, 908 – 914, 959 (Russian). · Zbl 0594.22001 [35] Silviu Teleman, Sur la représentation linéaire des groupes topologiques, Ann. Sci. Ecole Norm. Sup. (3) 74 (1957), 319 – 339 (French). · Zbl 0084.03106 [36] V. V. Uspenskiĭ, A universal topological group with a countable basis, Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 86 – 87 (Russian). · Zbl 0608.22003 [37] V. V. Uspenskij, The epimorphism problem for Hausdorff topological groups, Topology Appl. 57 (1994), no. 2-3, 287 – 294. · Zbl 0810.22002 [38] William A. Veech, Topological dynamics, Bull. Amer. Math. Soc. 83 (1977), no. 5, 775 – 830. · Zbl 0384.28018 [39] Jan de Vries, On the existence of \?-compactifications, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 3, 275 – 280 (English, with Russian summary). · Zbl 0378.54028 [40] J. de Vries, Elements of topological dynamics, Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. · Zbl 0783.54035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.