A class of II${}_1$ factors without non-trivial crossed product decompositions

Abstract

We introduce a class of separable II${}_1$ factors $M$ admitting no non-trivial crossed product decompositions: $M\ncong B{⋊}_{\sigma }G$, for any trace preserving action $G{\curvearrowright }^{\sigma }(B,\tau )$ of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau )$. These provide the first examples of II${}_1$ factors that do not arise as crossed products of noncommutative dynamical systems. Our approach relies on a novel construction of separable II${}_1$ factors $M$ whose embeddings into their tensor product square $M\overline{\otimes }M$ all arise from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$.

v2: minor changes