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On Hilbert dynamical systems

Published online by Cambridge University Press:  13 June 2011

ELI GLASNER  and
BENJAMIN WEISS
ELI GLASNER
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel (email: glasner@math.tau.ac.il)
BENJAMIN WEISS
Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel (email: weiss@math.huji.ac.il)
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Abstract

Returning to a classical question in harmonic analysis, we strengthen an old result of Walter Rudin. We show that there exists a weakly almost periodic function on the group of integers ℤ which is not in the norm-closure of the algebra B(ℤ) of Fourier–Stieltjes transforms of measures on the dual group , and which is recurrent. We also show that there is a Polish monothetic group which is reflexively but not Hilbert representable.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 629 - 642
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Akin, E., Auslander, J. and Berg, K.. When is a transitive map chaotic. Convergence in Ergodic Theory and Probability. Walter de Gruyter & Co., Berlin, New York, 1996, pp. 25–40.CrossRefGoogle Scholar
[2]Banaszczyk, W.. Additive Subgroups of Topological Vector Spaces (Lecture Notes in Mathematics, 1466). Springer, Berlin, 1991.CrossRefGoogle Scholar
[3]Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G.. Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar
[4]Downarowicz, T.. Weakly almost periodic flows and hidden eigenvalues. Topological Dynamics and Applications (Contemporary Mathematics, 215). American Mathematical Society, Providence, RI, 1998, pp. 101–120, a volume in honor of R. Ellis.CrossRefGoogle Scholar
[5]Dunkl, C. F. and Ramirez, D. E.. Topics in Harmonic Analysis. Appleton-Century-Crofts, New York, 1971.Google Scholar
[6]Ellis, R.. Distal transformation groups. Pacific J. Math. 8 (1957), 401–405.CrossRefGoogle Scholar
[7]Ellis, R. and Nerurkar, M.. Weakly almost periodic flows. Trans. Amer. Math. Soc. 313 (1989), 103–119.CrossRefGoogle Scholar
[8]Ferri, S. and Galindo, J.. Embedding a topological group into its WAP-compactification. Studia Math. 193 (2009), 99–108.CrossRefGoogle Scholar
[9]Galindo, J.. On unitary representability of topological groups. Math. Z. 263 (2009), 211–220.CrossRefGoogle Scholar
[10]Gao, S. and Pestov, V.. On a universality property of some abelian Polish groups. Fund. Math. 179 (2003), 1–15.CrossRefGoogle Scholar
[11]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[12]Glasner, E. and Maon, D.. Rigidity in topological dynamics. Ergod. Th. & Dynam. Sys. 9 (1989), 309–320.CrossRefGoogle Scholar
[13]Glasner, E. and Megrelishvili, M.. Linear representations of hereditarily non-sensitive dynamical systems. Colloq. Math. 104(2) (2006), 223–283.CrossRefGoogle Scholar
[14]Glasner, E. and Weiss, B.. Sensitive dependence on initial conditions. Nonlinearity 6 (1993), 1067–1075.CrossRefGoogle Scholar
[15]Glasner, E. and Weiss, B.. Locally equicontinuous dynamical systems. Colloq. Math. 84/85(Part 2) (2000), 345–361.CrossRefGoogle Scholar
[16]Host, B., Méla, J.-F. and Parreau, F.. Analyse harmonique des mesures. Astérisque 135–136 (1986), 261.Google Scholar
[17]Lemańczyk, M., Parreau, F. and Thouvenot, J.-P.. Gaussian automorphisms whose ergodic self-joinings are Gaussian. Fund. Math. 164 (2000), 253–293.CrossRefGoogle Scholar
[18]Megrelishvili, M.. Reflexively but not unitarily representable topological groups. Topology Proc. 25 (2000), 615–625, (2002).Google Scholar
[19]Megrelishvili, M.. Fragmentability and representations of flows. Topology Proc. 27:2 (2003), 497–544.Google Scholar
[20]Megrelishvili, M.. Reflexively but not Hilbert representable compact flows and semitopological semigroups. Colloq. Math. 110 (2008), 383–407.CrossRefGoogle Scholar
[21]Megrelishvili, M.. Topological transformation groups: selected topics. Open Problems in Topology II. Ed. Pearl, E.. Elsevier, Amsterdam, 2007, pp. 421–435.Google Scholar
[22]Pestov, V.. Dynamics of Infinite Dimensional Groups. The Ramsey–Dvoretzky–Milman Phenomenon (University Lecture Series, 40). American Mathematical Society, Providence, RI, 2006.Google Scholar
[23]Rudin, W.. Weak almost periodic functions and Fourier–Stieltjes transforms. Duke Math. J. 26 (1959), 215–220.CrossRefGoogle Scholar
[24]Ruppert, W.. Compact Semitopological Semigroups (Lecture Notes in Mathematics, 1079). Springer, Berlin, 1984.CrossRefGoogle Scholar
[25]Shtern, A.. Compact semitopological semigroups and reflexive representability. Russ. J. Math. Phys. 2 (1994), 131–132.Google Scholar