# Noncommutative Joinings II

## TL;DR

This paper extends the theory of noncommutative joinings by analyzing the relative independence of W*-dynamical systems, establishing conditions for disjointness relative to subsystems, and generalizing previous results from abelian groups to all locally compact groups.

## Contribution

It generalizes the concept of relative disjointness in noncommutative dynamical systems from abelian to all locally compact groups, providing a broader theoretical framework.

## Key findings

- Disjointness characterized by subsystem isomorphism.
- Generalization from abelian to all locally compact groups.
- Conditions for relative independence in W*-dynamical systems.

## Abstract

This paper is a continuation of the authors' previous work on noncommutative joinings, and contains a study of relative independence of W$^*$-dynamical systems. We prove that, given any separable locally compact group $G$, an ergodic W$^{*}$-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.