On Hilbert dynamical systems (Q2908149)

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.