# On Noncommutative Joinings

## TL;DR

This paper generalizes classical joining theory of dynamical systems to the noncommutative setting, establishing new results and characterizations using quantum channels and operator algebras.

## Contribution

It introduces a noncommutative framework for joinings, linking them with equivariant quantum channels, and extends classical properties like ergodicity and mixing to this setting.

## Key findings

- Joinings are identified with equivariant quantum channels.
- Classical disjointness characterizations are extended to noncommutative systems.
- Analogues of ergodicity, primeness, and mixing are established in the noncommutative context.

## Abstract

This paper extends the classical theory of joinings of measurable dynamical systems to the noncommutative setting from several interconnected points of view. Among these is a particularly fruitful identification of joinings with equivariant quantum channels between $W^{\ast}$-dynamical systems that provides noncommutative generalizations of many fundamental results of classical joining theory. We obtain fully general analogues of the main classical disjointness characterizations of ergodicity, primeness and mixing phenomena.