Let $\Cset[G]$ be the group ring of a discrete group $G$ . It has two completions to a $C^*$ -algebra:

Reduced group $C^*$ -algebra.

The reduced group $C^*$ -algebra, $C^*_r(G)$ , is obtained by completing $\Cset[G]$ in the operator norm for its regular representation on $l^2(G)$ .

Maximal group $C^*$ -algebra.

The maximal group $C^*$ -algebra, $C^*_\mathrm{max}(G)$ or just $C^*(G)$ , is defined by the following universal property: any *-homomorphism from $\Cset[G]$ to some $\Bset(\hilbert)$ (the $C^*$ -algebra of bounded operators on some Hilbert space $\hilbert$ ) factors through the inclusion $\Cset[G] \hookrightarrow C^*_\mathrm{max}(G)$ .

If $G$ is amenable then $C^*_r(G) \cong C^*_\mathrm{max}(G)$ .