Model theory of a non-degenerate representation of a unital C*-algebra

TL;DR

This paper develops a continuous logic framework for non-degenerate representations of unital C*-algebras on Hilbert spaces, providing axiomatizations, type characterizations, and identifying the model companion of the theory.

Contribution

It offers an explicit axiomatization and a detailed analysis of the model theory of non-degenerate C*-algebra representations, including quantifier elimination and model companion characterization.

Findings

Types correspond to positive linear functionals over A.

The theory admits quantifier elimination.

The model companion of the theory is characterized.

Abstract

We study the theory of a Hilbert space H as a module for a unital C*-algebra 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. We show that for every v in H the type of v over the empmtyset is in correspondence with the positive linear functional over A defined by v and has quantifier elimination as well. Finally, we characterize the model companion of the incomplete theory of all non-degenerate representations of A.