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Banach Representations and Affine Compactifications of Dynamical Systems

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Asymptotic Geometric Analysis

Part of the book series: Fields Institute Communications ((FIC,volume 68))

Abstract

To every Banach space V we associate a compact right topological affine semigroup ℰ(V ). We show that a separable Banach space V is Asplund if and only if \(\mathcal{E}(V )\) is metrizable, and it is Rosenthal (i.e., it does not contain an isomorphic copy of l 1) if and only if \(\mathcal{E}(V )\) is a Rosenthal compactum. We study representations of compact right topological semigroups in \(\mathcal{E}(V )\). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily nonsensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility.

Mathematical Subject Classifications (2010): 37Bxx, 54H20, 54H15, 46xx

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Authors and Affiliations

  1. Department of Mathematics, Tel-Aviv University, Ramat Aviv, Tel Aviv, Israel

    Eli Glasner

  2. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Michael Megrelishvili

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  1. Eli Glasner
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  2. Michael Megrelishvili
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Correspondence to Michael Megrelishvili .

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Editors and Affiliations

  1. TU Wien, Wiedner Hauptstrasse 8–10, Wien, 1040, Austria

    Monika Ludwig

  2. Fac. Exact Sciences, School of Mathematical Sciences, University of Tel Aviv, Schreiber Building 334, Tel Aviv, 69978, Israel

    Vitali D. Milman

  3. University of Ottawa, Ottawa, K1N 6N5, Canada

    Vladimir Pestov

  4. , Department of Math and Stat Scis, University of Alberta, Central Academic Bldg 632, Edmonton, T6G 2G1, Alberta, Canada

    Nicole Tomczak-Jaegermann

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Glasner, E., Megrelishvili, M. (2013). Banach Representations and Affine Compactifications of Dynamical Systems. In: Ludwig, M., Milman, V., Pestov, V., Tomczak-Jaegermann, N. (eds) Asymptotic Geometric Analysis. Fields Institute Communications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6406-8_6

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