---
tag: 8HG8TS46A6
kind: concept
name: Crossed product
msc: [46L55, 46L05, 47L65]
sources:
  - url: https://en.wikipedia.org/wiki/Crossed_product
---

## Summary

The crossed product $A \rtimes_\alpha G$ is a C*-algebra built from a C\*-algebra $A$ and a group $G$.
The group acts on $A$ by automorphisms through a [[group action|0WCY42EV9V]] $\alpha$.
When $A$ is unital and $G$ is discrete, it contains $A$ and unitaries $u_g$ for $g \in G$.
In it the action becomes conjugation: $u_g a u_g^* = \alpha_g(a)$.
It turns a dynamical system into a single algebra, and many examples, such as the noncommutative tori, arise this way.
There is a parallel construction for von Neumann algebras.

## Explanations

### L1 human

Suppose a group moves the points of a space around.
The crossed product puts the functions on the space and the group elements into one algebra.
Moving a function by the group then becomes the algebraic operation $a \mapsto u_g a u_g^{*}$.
For the group $\Z$ rotating a circle by an irrational angle, the crossed product is the noncommutative torus.