Eberlein oligomorphic groups

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 ℵ 0 -stable, ℵ 0categorical structures. This analysis is then extended to all semitopological semigroup compactifica-2010 Mathematics Subject Classification. Primary: 22F50. Secondary: 03C45, 22A25, 03C98, 22A15.

Infinite-dimensional Polish groups and Property (T)

“…For some reason, this «semi-global» notion of independence is hardly ever presented or considered in this way in the model-theoretic literature. It can however be very useful in the context of applications, as has already been shown in the works [BIT18] and [IT18].…”

“…The new characterization of PRP groups opened the door for novel interactions with model-theory. A precise dictionary between several topologicaldynamic features of PRP groups and model-theoretic properties of the associated structures arose from the works [BT16,Iba16,BIT18], along with several applications.…”

Automorphism Groups of Randomized Structures

“…Thus, an interesting stronger property is satisfied if the space V can be chosen to be a Hilbert space. For the case of R(G), this property is therefore a strengthening of stability, and has been investigated as such in [11]. We showed there that, for a classical ℵ 0 -categorical structure M , R(G) is a Hilbert-representable semitopological semigroup if and only if M is stable and one-based (equivalently, ℵ 0 -stable).…”

“…For ℵ 0 -categorical structures we have the following pleasant fact, observed in [ Proof. See [11], Fact 2.14. The proof adapts readily to the case of metric structures, as per [13], Lemma 2.3.…”

“…If M is the separable model of an ℵ 0 -categorical theory T , then most of the model-theoretic information of T is coded by dynamical properties of G = Aut(M ), as per the recent works [11,13,23]. On the other hand, the randomized theory T R is also ℵ 0 -categorical; hence, in this case, the Borel randomization encompasses all the model-theoretic information of T R , and this can be recovered from the group G Ω.…”