Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products

We develop a relative boundary theory for actions of discrete groups on compact spaces and use it to derive rigidity results for reduced crossed products. For a discrete group $\Gamma$ acting on a compact space $X$ and a subgroup $H$, we construct a universal boundary over $X$ which is minimal as a $\Gamma$-system and strongly proximal with respect to $H$. When $H\le_c\Gamma$ is commensurated and the $H$-action on $X$ is minimal, we show that this universal boundary agrees, in a canonical $\Gamma$-equivariant way, with the generalized Furstenberg boundary of $(H,X)$, thereby unifying and extending earlier results on relative boundaries. As an application, we introduce the notion of an $X$-plump subgroup given a $\Gamma$-space $X$, a generalized version of plumpness tailored to crossed products. Under natural dynamical hypotheses, this leads to new examples of irreducible $C^*$-inclusions. Under additional assumptions, we also show that every intermediate $C^*$-algebra is a crossed product.