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 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 ∈ H the type tp(v/∅) 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.

FORKING AND STABILITY IN THE REPRESENTATIONS OF A C*-ALGEBRA

“…As we said before, this is a continuation of a previous work (see [3]). The main results obtained there are the following:…”

“…Gelfand-Naimark-Segal construction is a tool for understanding definable closures (see Theorem 3.11 in [3]).…”

“…In this section we give a characterization of definable and algebraic closures. Gelfand-Naimark-Segal construction is a tool for understanding definable closures (see Theorem 3.11 in [3]).…”

Continuous theory of operator expansions of finite dimensional Hilbert spaces and decidability

“…Finishing this section we mention papers [5, 8, 13, 16, 19] where modern applications of continuous logic to functional analysis were initiated. Our approach is similar to approaches of Argoty, Berenstein and Henson presented in particular in [1, 2, 9].…”

“…As a result we see that is a ‐module. We now apply [1, Theorem 2.20]. It states that two representations of a ‐algebra are elementarily equivalent in continuous logic if and only if for any the ranks of the corresponding elements are finite and the same or are infinite.…”