Group C*-algebra

Summary

The group C*-algebra $C^*(G)$ of a locally compact group $G$ is a completion of the group algebra $L^1(G)$. It uses the largest norm coming from unitary representations of $G$. It is a C*-algebra whose nondegenerate representations correspond exactly to the unitary representations of $G$. The reduced group C*-algebra $C^*_r(G)$ uses the left regular representation alone. These algebras let one apply the theory of operator algebras to groups, and they are central in noncommutative geometry and the Baum–Connes conjecture.