Model theory of Hilbert spaces with a discrete group action

In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a discrete countable group. When the group is finite, we prove the theory of the corresponding expansion, regardless if it is existentially closed, has quantifier elimination, is ℵ 0 -categorical, ℵ 0 -stable and SFB. On the other hand, when the group involved is countably infinite, the theory of the Hilbert space expanded by the representation of this group is ℵ 0 -categorical up to perturbations. Additionally, when the expansion is model complete, we prove that it is ℵ 0 -stable up to perturbations.