nach oben

Erschienen in:

2013 | OriginalPaper | Buchkapitel

Banach Representations and Affine Compactifications of Dynamical Systems

verfasst von : Eli Glasner, Michael Megrelishvili

Erschienen in: Asymptotic Geometric Analysis

Verlag: Springer New York

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel On a Stability Property of the Generalized Spherical Radon Transform

Nächstes Kapitel Flag Measures for Convex Bodies

E. Akin, E. Glasner, Residual properties and almost equicontinuity. J. d’Anal. Math. 84, 243–286 (2001)MathSciNetMATHCrossRef

E.M. Alfsen, Compact Convex Sets and Boundary Integrals (Springer, Berlin, 1971)MATHCrossRef

L. Asimow, A.J. Ellis, Convexity Theory and Its Applications in Functional Analysis (Academic, New York, 1980)MATH

R.N. Ball, J.N. Hagler, Real-valued functions on flows. Preprint (1996)

W. Bauer, K. Sigmund, Topological dynamics of transformations induced on the space of probability measures. Monatsh. für math. 79, 81–92 (1975)MathSciNetMATHCrossRef

J.F. Berglund, H.D. Junghenn, P. Milnes, Compact Right Topological Semigroups and Generalizations of Almost Periodicity. Lecture Notes in Mathematics, vol. 663 (Springer, Berlin, 1978)

J.F. Berglund, H.D. Junghenn, P. Milnes, Analysis on Semigroups (Wiley, New York, 1989)MATH

K. Berg, D. Gove, K. Haddad, Enveloping semigroups and mappings onto the two-shift. Proc. Am. Math. Soc. 126, 899–905 (1998)MathSciNetMATHCrossRef

C. Bessaga, A. Pelczynski, Selected Topics in Infinite-Dimensional Topology (Monografie Matematyczne, Warszawa, 1975)MATH

10.

J. Bourgain, D.H. Fremlin, M. Talagrand, Pointwise compact sets in Baire-measurable functions. Am. J. Math. 100(4), 845–886 (1977)MathSciNetCrossRef

11.

M. Boyle, Lower entropy factors on topological dynamics. Ergod. Theor. Dyn. Syst. 4, 541–557 (1984)MathSciNetCrossRef

12.

G. Choquet, Lectures on Analysis, vol. II (W. A. Benjamin, New York–Amsterdam, 1969)

13.

H.G. Dales, A.T. Lau, D. Strauss, Banach algebras on semigroups and on their compactifications, Memoir. Am. Math. Soc. 205, no. 966, pp. 165 (2010)

14.

W.J. Davis, T. Figiel, W.B. Johnson, A. Pelczyński, Factoring weakly compact operators. J. Funct. Anal. 17, 311–327 (1974)MATHCrossRef

15.

K. de Leeuw, I. Glicksberg, Applications of almost periodic compactifications. Acta. Math. 105, 63–97 (1961)MathSciNetMATHCrossRef

16.

R. Ellis, Lectures on Topological Dynamics (W. A. Benjamin Inc., New York, 1969)MATH

17.

M. Fabian, Gateaux Differentiability of Convex Functions and Topology: Weak Asplund Spaces. Canadian Mathematical Society Series of Monographs and Advanced Texts (Wiley, New York, 1997)

18.

I.M. Gelfand, A.N. Kolmogoroff, On the rings of continuous functions on topological spaces. Doklady Acad. Sci. USSR 22, 11–15 (1939)

19.

E. Glasner, Distal and semisimple affine flows. Am. J. Math. 109, 115–132 (1987)MathSciNetMATHCrossRef

20.

E. Glasner, Ergodic Theory Via Joinings. Mathematical Surveys and Monographs, vol. 101 (AMS, Providence, 2003)

21.

E. Glasner, On tame dynamical systems. Colloq. Math. 105, 283–295 (2006)MathSciNetMATHCrossRef

22.

E. Glasner, The structure of tame minimal dynamical systems. Ergod. Theor. Dyn. Syst. 27, 1819–1837 (2007)MathSciNetMATHCrossRef

23.

E. Glasner, D. Maon, Rigidity in topological dynamics. Ergod. Theor. Dyn. Syst. 9, 309–320 (1989)MathSciNetMATHCrossRef

24.

E. Glasner, M. Megrelishvili, Linear representations of hereditarily non-sensitive dynamical systems. Colloq. Math. 104(2), 223–283 (2006)MathSciNetMATHCrossRef

25.

E. Glasner, M. Megrelishvili, New algebras of functions on topological groups arising from G-spaces. Fund. Math. 201, 1–51 (2008)MathSciNetMATHCrossRef

26.

E. Glasner, M. Megrelishvili, On fixed point theorems and nonsensitivity. Isr. J. Math. 190, 289–305 (2012)MathSciNetMATHCrossRef

27.

E. Glasner, M. Megrelishvili, Representations of dynamical systems on Banach spaces not containing l ₁. Trans. Am. Math. Soc. 364, 6395–6424 (2012)MathSciNetCrossRef

28.

E. Glasner, M. Megrelishvili, Eventual nonsensitivity and tame dynamical systems. In preparation

29.

E. Glasner, M. Megrelishvili, V.V. Uspenskij, On metrizable enveloping semigroups. Isr. J. Math. 164, 317–332 (2008)MathSciNetMATHCrossRef

30.

E. Glasner, B. Weiss, Quasifactors of zero-entropy systems. J. Am. Math. Soc. 8, 665–686 (1995)MathSciNetMATH

31.

E. Glasner, B. Weiss, Hilbert Dynamical Systems. Ergodic Theory Dynam. Systems 32(2), 629–642 (2012)MathSciNetMATHCrossRef

32.

G. Godefroy, Compacts de Rosenthal. Pacific J. Math. 91, 293–306 (1980)MathSciNetMATHCrossRef

33.

R. Haydon, Some new characterizations of Banach spaces containing l ₁. Math. Proc. Camb. Phil. Soc. 80, 269–276 (1976)MathSciNetMATHCrossRef

34.

W. Huang, Tame systems and scrambled pairs under an abelian group action. Ergod. Theor. Dyn. Syst. 26, 1549–1567 (2006)MATHCrossRef

35.

S. Immervoll, A note on the enveloping semigroup of a flow. Proc. Roy. Irish Acad. 99, 185–188 (1999)MathSciNet

36.

A. Jalilian, M.A. Pourabdolah, Universal compactifications of transformation semigroups. J. Sci. I.R. Iran 11(3), 239–244 (2000)

37.

J.E. Jayne, J. Orihuela, A.J. Pallares, G. Vera, σ-fragmentability of multivalued maps and selection theorems. J. Funct. Anal. 117(2), 243–273 (1993)

38.

J.E. Jayne, C.A. Rogers, Borel selectors for upper semicontinuous set-valued maps. Acta. Math. 155, 41–79 (1985)MathSciNetMATHCrossRef

39.

H.D. Junghenn, Operator semigroup compactifications. Trans. Am. Math. Soc. 348, 1051–1073 (1996)MathSciNetMATHCrossRef

40.

D. Kerr, H. Li, Independence in topological and C ^∗-dynamics. Math. Ann. 338, 869–926 (2007)MathSciNetMATHCrossRef

41.

A. Köhler, Enveloping semigrops for flows. Proc. Roy. Irish Acad. 95A, 179–191 (1995)

42.

J.D. Lawson, Joint continuity in semitopological semigroups. Illinois J. Math. 18, 275–285 (1974)MathSciNetMATH

43.

J.D. Lawson, Additional notes on continuity in semitopological semigroups. Semigroup Forum 12, 265–280 (1976)MathSciNetMATHCrossRef

44.

A. Lima, O. Nygaard, E. Oja, Isometric factorization of weakly compact operators and the approximation property. Isr. J. Math. 119, 325–348 (2000)MathSciNetMATHCrossRef

45.

M. Megrelishvili, Fragmentability and continuity of semigroup actions. Semigroup Forum 57, 101–126 (1998)MathSciNetMATHCrossRef

46.

M. Megrelishvili, Every semitopological semigroup compactification of the group H ₊[0, 1] is trivial. Semigroup Forum 63(3), 357–370 (2001)MathSciNetMATHCrossRef

47.

M. Megrelishvili, Operator topologies and reflexive representability, in Nuclear groups and Lie groups. Research and Exposition in Mathematical series, vol. 24 (Heldermann, Berlin, 2001), pp. 197–208

48.

M. Megrelishvili, Fragmentability and representations of flows. Topology Proc. 27(2), 497–544 (2003). See also: www.math.biu.ac.il/~megereli.

49.

M. Megrelishvili, Compactifications of semigroups and semigroup actions. Topology Proc. 31(2), 611–650 (2007)MathSciNet

50.

M. Megrelishvili, Reflexively representable but not Hilbert representable compact flows and semitopological semigroups. Colloq. Math. 110, 383–407 (2008)MathSciNetMATHCrossRef

51.

M. Megrelishvili, V. Pestov, V. Uspenskij, A note on the precompactness of weakly almost periodic groups, in Nuclear Groups and Lie Groups. Research and Exposition in Mathematic Series, vol. 24 (Heldermann, Lemgo, 2001), pp. 209–216

52.

W.B. Moors, I. Namioka, Furstenberg’s structure theorem via CHART groups. Ergod. Theor. Dyn. Syst. Available on CJO 2012. doi:10.1017/ S0143385712000089

53.

I. Namioka, Right topological groups, distal flows and a fixed point theorem. Math. Syst. Theor. 6, 193–209 (1972)MathSciNetMATHCrossRef

54.

I. Namioka, Affine flows and distal points. Math. Z. 184, 259–269 (1983)MathSciNetMATHCrossRef

55.

I. Namioka, Radon-Nikodým compact spaces and fragmentability. Mathematika 34, 258–281 (1987)MathSciNetMATHCrossRef

56.

I. Namioka, R.R. Phelps, Banach spaces which are Asplund spaces. Duke Math. J. 42, 735–750 (1975)MathSciNetMATHCrossRef

57.

D. Penazzi, An extension to the non-metric case of a theorem of Glasner. Rend. Sem. Mat. Univ. Pol. Torino 59, 37–41 (2001)MathSciNetMATH

58.

V. Pestov, Topological groups: where to from here? Topology Proc. 24, 421–502 (1999). http://arXiv.org/abs/math.GN/9910144

59.

R.R. Phelps, Lectures on Choquet’s Theorem (D. Van Nostrand Company, Princeton, 1966)MATH

60.

J.S. Pym, Compact semigroups with one sided continuity, in The Analytical and Topological Theory of Semigroups, ed. by K.H. Hofmann, J.D. Lawson, J.S. Pym. (Walter de grueter, Berlin, 1990), pp. 197–217

61.

J.S. Pym, Affine semigroups and second duals. Semigroup Forum 47, 130–132 (1993)MathSciNetMATHCrossRef

62.

H.P. Rosenthal, A characterization of Banach spaces containing l ₁. Proc. Nat. Acad. Sci. USA 71, 2411–2413 (1974)MATHCrossRef

63.

W. Ruppert, Rechtstopologische Halbgruppen. J. Reine Angew. Math. 261, 123–133 (1973)MathSciNetMATH

64.

W. Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory. Lecture Notes in Mathematics, vol. 1079 (Springer, New York, 1984)

65.

E. Saab, P. Saab, A dual geometric characterization of Bancah spaces not containing l ₁. Pacific J. Math. 105(2), 413–425 (1983)MathSciNetCrossRef

66.

Z. Semadeni, Banach Spaces of Continuous Functions, vol. 1 (PWN Polish Scientific Publishers, Warsaw, 1971)MATH

67.

A. Shtern, Compact semitopological semigroups and reflexive representability. Russian J. Math. Physics 2, 131–132 (1994)MathSciNetMATH

68.

M. Talagrand, Pettis integral and measure theory. Memoir. Am. Math. Soc. 51, no. 307, pp. 224 (1984)

69.

S. Todorc̆ević, Topics in Topology. Lecture Notes in Mathematics, vol. 1652 (Springer, New York, 1997)

70.

V.V. Uspenskij, Compactifications of topological groups. Proceedings of the Ninth Prague Topological Symposium (Prague, 19–25 August 2001), ed. by P. Simon, Topology Atlas (electronic publication), Toronto, April 2002, pp. 331–346. arXiv:math.GN/0204144

71.

J. de Vries, Equivariant embeddings of G-spaces, in General Topology and its Relations to Modern Analysis and Algebra IV, Part B, ed. J. Novak, Prague, 1977, pp. 485–493

72.

J. de Vries, Elements of Topological Dynamics (Kluwer Academic Publishers, Dordrecht, 1993)MATHCrossRef

73.

S. Williams, Toeplitz minimal flows which are not uniquely ergodic. Z. Wahrsch. Verw. 67, 95–107 (1984)MATHCrossRef

74.

K. Witz, Applications of a compactification for bounded operator semigroups. Illinois J. Math. 8, 685–696 (1964)MathSciNetMATH

Titel: Banach Representations and Affine Compactifications of Dynamical Systems
verfasst von: Eli Glasner
Michael Megrelishvili
Verlag: Springer New York
Buch: Asymptotic Geometric Analysis
Print ISBN: 978-1-4614-6405-1

Electronic ISBN: 978-1-4614-6406-8

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-6406-8_6

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"