Model theory of Hilbert spaces expanded by a representation of a group

In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a group. When the group is finite, we prove the theory of the corresponding expansion is $\aleph_0$-categorical, $\aleph_0$-stable and is SFB. On the other hand, when the group involved is a pro...

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Bibliographic Details
Main Authors Berenstein, Alexander, Pérez, Juan Manuel
Format Journal Article
LanguageEnglish
Published 05.09.2024
Subjects
Mathematics - Logic
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Summary:In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a group. When the group is finite, we prove the theory of the corresponding expansion is $\aleph_0$-categorical, $\aleph_0$-stable and is SFB. On the other hand, when the group involved is a product of the form $H\times \mathbb{Z}^n$, where $H$ is a finite group and $n\geq 1$, the theory of the Hilbert space expanded by the representation of this group is, in general, stable not $\aleph_0$-stable, not $\aleph_0$-categorical, but it is $\aleph_0$-categorical up to perturbations and $\aleph_0$-stable up to perturbations.
DOI:10.48550/arxiv.2409.03923