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

We introduce a class of separable II$_1$ factors $M$ admitting no non-trivial crossed product decompositions: $M\not\cong B\rtimes_\sigma G$, 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 $x\mapsto 1\otimes x$.