An exotic II${}_1$ factor without property Gamma

Abstract

We introduce a new iterative amalgamated free product construction of II${}_1$ factors, and use it to construct a separable II${}_1$ factor which does not have property Gamma and is not elementarily equivalent to the free group factor $\text{L}({\mathbb{F}}_n)$, for any $2\le n\le \infty$. This provides the first explicit example of two non-elementarily equivalent II${}_1$ factors without property Gamma. Moreover, our construction also provides the first explicit example of a II${}_1$ factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu's free entropy theory and Popa's deformation/rigidity theory.

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