On Hilbert dynamical systems

TL;DR

This paper explores properties of weakly almost periodic functions and their relation to Fourier-Stieltjes transforms, revealing new distinctions in harmonic analysis and group representations.

Contribution

It strengthens Rudin's classical result by constructing a recurrent weakly almost periodic function outside the norm-closure of B(Z) and identifies a Polish monothetic group that is reflexively but not Hilbert representable.

Findings

Existence of a recurrent weakly almost periodic function not in the norm-closure of B(Z)

Construction of a Polish monothetic group reflexively but not Hilbert representable

Enhanced understanding of harmonic analysis on groups

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 circle, the dual group of Z, and which is recurrent. We also show that there is a Polish monothetic group which is reflexively but not Hilbert representable.