On Hilbert dynamical systems
type: Item
instance of: scholarly article
title: On Hilbert dynamical systems
zbMATH Open document ID: 1248.37019
DOI: 10.1017/S0143385711000277
author: Benjamin Weiss
author: Eli Glasner
published in: Ergodic Theory and Dynamical Systems
publication date: 2012-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.0230
Mathematics Subject Classification ID: 37B20
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 6076544
zbMATH Keywords: weakly almost periodic functions
zbMATH Keywords: recurrent functions
zbMATH Keywords: Polish monothetic group
MaRDI profile type: Publication
arXiv ID: 1005.0230
cites work: On unitary representability of topological groups
cites work: Sensitive dependence on initial conditions
cites work: Embedding a topological group into its WAP-compactification
cites work: Hereditarily non-sensitive dynamical systems and linear representations
cites work: Rigidity in topological dynamics
cites work: Distral transformation groups
cites work: Q4800032
cites work: On a universality property of some abelian Polish groups
cites work: Additive subgroups of topological vector spaces
cites work: Compact semitopological semigroups: An intrinsic theory
cites work: Weak almost periodic functions and Fourier-Stieltjes transforms
cites work: Q3415741
cites work: Reflexively representable but not Hilbert representable compact flows and semitopological semigroups
OpenAlex ID: W2963467280
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review text: In [Duke Math. J. 26, 215--220 (1959; Zbl 0085.32502)], \textit{W. Rudin} showed that the norm-closure \(H(\mathbb{Z}) = \overline{B(\mathbb{Z})}\) of the algebra \(B(\mathbb{Z})\) of Fourier-Stieltjes transforms of complex measures on \(\hat{\mathbb{Z}} = \mathbb{T}\) is strictly contained in the algebra \(\mathrm{WAP}(\mathbb{Z})\) of weakly almost periodic functions on \(\mathbb{Z}\). All examples of functions in \(\mathrm{WAP}(\mathbb{Z})\setminus H(\mathbb{Z})\) known prior to this article are non-recurrent. It is a natural question to ask whether \(\mathrm{WAP}(\mathbb{Z})\setminus H(\mathbb{Z})\) contains recurrent functions as well. This article establishes the existence of many recurrent functions in \(\mathrm{WAP}(\mathbb{Z})\setminus H(\mathbb{Z})\). Moreover, it is shown that the question on the existence of a Polish monothetic group which can be represented as a group of isometries of a reflexive Banach space but cannot be represented as a group of unitary operators on a Hilbert space is closely related. Also this question is answered affirmatively. The proofs use the theory of topological dynamical systems.
reviewed by: Anke D. Pohl
description: scientific article; zbMATH DE number 6076544