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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -categorical, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -categorical up to perturbations. Additionally, when the expansion is model complete, we prove that it is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -stable up to perturbations.