in a joint work with d. hoff and a. ioana, we have discovered the following product rigidity phenomenon: if $\gamma$ is an icc group measure equivalent to a product of non-elementary hyperbolic groups, then any tensor product decomposition of the ii$_1$ factor $l(\gamma)$ arises only from the canonical direct product decomposition of $\gamma$. subsequently, i. chifan, r. de santiago and w. sucpikarnin classified all the tensor product decompositions for group von neumann algebras arising from a large class of amalgamated free products. in this talk we will give an overview of these results and discuss about a similar rigidity phenomenon that appears in the context of von neumann algebras arising from actions. more precisely, we prove that if $\gamma$ is a product of certain groups and $\gamma\curvearrowright (x,\mu)$ is an arbitrary free ergodic measure preserving action, then we show that any tensor product decomposition of the ii$_1$ factor $l^\infty(x)\rtimes\gamma$ arises only from the canonical direct product decomposition of the underlying action $\gamma\curvearrowright x.$