group $C^{*}$-algebra

Let $ℂ[G]$ 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 $ℂ[G]$in the operator norm for its regular representation on $l^2(G)$.

Maximal group $C^{*}$-algebra.

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

If $G$ is amenable then $C_r^{*}(G)\cong C_{\max }^{*}(G)$.