Noncommutative Joinings II

This paper is a continuation of the authors’ previous work on noncommutative joinings, and contains a study of relative independence of W${}^{\ast }$-dynamical systems. We prove that, given any separable locally compact group $G$, an ergodic W${}^{\ast }$-dynamical $G$-system $\mathfrak{M}$ with compact subsystem $\mathfrak{N}$ is disjoint relative to $\mathfrak{N}$ from its maximal compact subsystem ${\mathfrak{M}}_K$ if and only if $\mathfrak{N}\cong {\mathfrak{M}}_K$. This generalizes recent work of Duvenhage, which established the result for $G$ abelian.