Eberlein oligomorphic groups

TL;DR

This paper explores the Fourier--Stieltjes algebra of certain Polish groups, linking their algebraic properties to model-theoretic structures and characterizing their compactifications and representations.

Contribution

It provides a model-theoretic description of the Hilbert compactification for Roelcke precompact groups and characterizes when their Fourier--Stieltjes algebra is dense in weakly almost periodic functions.

Findings

Automorphism groups of -stable, -categorical structures have dense Fourier--Stieltjes algebra.

Semitopological semigroup compactifications are Hilbert-representable iff they are inverse semigroups.

Every factor of the Hilbert compactification is Hilbert-representable.

Abstract

We study the Fourier--Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups whose Fourier--Stieltjes algebra is dense in the algebra of weakly almost periodic functions: those are exactly the automorphism groups of $\aleph_0$-stable, $\aleph_0$-categorical structures. This analysis is then extended to all semitopological semigroup compactifications $S$ of such a group: $S$ is Hilbert-representable if and only if it is an inverse semigroup. We also show that every factor of the Hilbert compactification is Hilbert-representable.