The model theory of modules of a C*-algebra

Abstract

We study the theory of a Hilbert space H as a module for a unital C*-algebra $${\mathcal{A}}$$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of $${\mathcal{A}}$$ . Finally, we show that there is an homeomorphism between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra $${\mathcal{A}}$$ .