# Enforceable operator algebras

## TL;DR

This paper extends game-based model construction to continuous logic, exploring enforceability of operator algebras, and connects the Connes Embedding Problem with the enforceability of the hyperfinite II$_1$ factor.

## Contribution

It introduces a framework for enforceable models in continuous logic and applies it to operator algebras, linking enforceability to major open problems.

## Key findings

- Hyperfinite II$_1$ factor is enforceable iff Connes Embedding Problem is positive.
- The continuous functions on the pseudoarc form an enforceable and prime model.
- Establishes a connection between enforceability and classical open problems in 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 \cstar-algebras and use this to show that it is the prime model of its theory.