Group algebra of a locally compact group

Summary

The group algebra of a locally compact group $G$ is the space $L^1(G)$ of integrable functions on $G$. Its product is convolution, and it has a natural involution. It is a Banach $*$-algebra, and it plays for $G$ the role that the finite group algebra plays for a finite group. Unitary representations of $G$ correspond to nondegenerate $*$-representations of $L^1(G)$, so the representation theory of $G$ becomes the representation theory of one algebra.