Existentially closed II_1 factors

TL;DR

This paper investigates the properties of existentially closed II_1 factors, demonstrating their automorphism characteristics, model completeness issues, and the existence of many nonisomorphic models, advancing understanding in operator algebra theory.

Contribution

It proves that Th(R) is not model-complete and establishes the existence of continuum many nonisomorphic existentially closed models of Th(R).

Findings

Automorphisms of existentially closed II_1 factors are approximately inner.

Th(R) is not model-complete.

There are continuum many nonisomorphic existentially closed models of Th(R).

Abstract

We examine the properties of existentially closed (R^omega-embeddable) II_1 factors. In particular, we use the fact that every automorphism of an existentially closed (R^omega-embeddable) II_1 factor is approximately inner to prove that Th(R) is not model-complete. We also show that Th(R) is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th(R).