Enforceable Operator Algebras

Abstract

We adapt the classical notion of building models by games to the setting of continuous model theory. As an application, we study to what extent canonical operator algebras are enforceable models. For example, we show that the hyperfinite II $_{1}$ factor is an enforceable II $_{1}$ factor if and only if the Connes Embedding Problem has a positive solution. We also show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian $\text{C}^{\ast }$ -algebras and use this to show that it is the prime model of its theory.

An Invitation to Model Theory and C*-Algebras

“…Let us now formally introduce infinitary formulas in an arbitrary language L as in the framework considered in Section 2.3. In fact, we will only consider a special case of infinitary formulas, which we call sup inf-formulas following [53]. Recall that, if ϕ is a finitary formula (or, more generally, a definable predicate) and M is an L-structure, then the interpretation ϕ M of ϕ in M is a uniformly continuous function.…”

“…Using the techniques of model-theoretic forcing and building models by games, many deep open problems in operator algebra theory have been reformulated in terms of model-theoretic notions, in the hope that these might be more amenable to a direct attack. Examples of these problems include the famous Connes Embedding Problem and some its C*-algebraic counterparts (the Kirchberg Embedding Problem, the MF problem); see [36,53,55,56].…”

“…The modeltheoretic proof of the characterization of D-absorption presented here is in some respects original, although heavily inspired by the proofs from the literature [47,75,83]. Finally, the technique of model-theoretic forcing in the metric setting has been first considered in [12] and then further developed in [36,42,53]. §2.…”

$\mathcal{R}$ We Living in the Matrix?

“…We define the space The ultraproduct of ( , ) ∞ =1 along is the II 1 factor ∏ / = ℓ ∞ ( )/ℐ with trace ( ) = lim → ( ). It is a nontrivial exercise to show that the ultraproduct of II 1 factors is again a II 1 factor (see [1,4]). If = for every ∈ ℕ, then we write ∏ / ∶= , and in this case we say that is the ultrapower of .…”

“…The interplay between the operator norm and the 2-norm on introduces complexity when considering as a logical structure. We remark that the understanding of continuous model theory of II 1 factors is currently in its infancy (see [4,5]).…”