Intermediate Factor theorems are central to many rigidity results, starting from Celebrated Margulis's Normal Subgroup Theorem to the well-known Nevo-Stuck-Zimmer Theorem. In this talk, we will discuss a non-commutative generalization of it. In particular, we will prove a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given a lattice $\Gamma < G_1 \times \dots \times G_d$ with dense projections and a trace-preserving ergodic action $\Gamma \curvearrowright (\mathcal{N}, \tau)$, we show that every intermediate von Neumann algebra between $\mathcal{N}\rtimes\Gamma$ and $(L^\infty(B,\nu_B)\overline{\otimes}\mathcal{N})\rtimes\Gamma$ is again a crossed product. This is a recent joint work with Yongle Jiang and Shuoxing Zhou.

No prior knowledge about these objects will be assumed. I will take the first half to explain these objects in detail and trace the history a bit.