Existentially closed ${\rm II}_1$ factors
Volume 233 / 2016
Fundamenta Mathematicae 233 (2016), 173-196
MSC: Primary 03C07; Secondary 03C50, 03C25, 46L36.
DOI: 10.4064/fm126-12-2015
Published online: 11 December 2015
Abstract
We examine the properties of existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm II}_1$ factors. In particular, we use the fact that every automorphism of an existentially closed ($\mathcal {R}^\omega $-embeddable) ${\rm II}_1$ factor is approximately inner to prove that $\operatorname {Th}(\mathcal {R})$ is not model-complete. We also show that $\operatorname {Th}(\mathcal {R})$ 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 $\operatorname {Th}(\mathcal {R})$.
