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 $\alpha$. 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.