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group $C^{*}$algebra
Let $\mathbb{C}[G]$ 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 $\mathbb{C}[G]$ 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 $\mathbb{C}[G]$ to some $\mathbb{B}(\mathord{\mathcal{H}})$ (the $C^{*}$algebra of bounded operators on some Hilbert space $\mathord{\mathcal{H}}$) factors through the inclusion $\mathbb{C}[G]\hookrightarrow C^{*}_{\mathrm{max}}(G)$.
If $G$ is amenable then $C^{*}_{r}(G)\cong C^{*}_{\mathrm{max}}(G)$.
Related:
CAlgebra, GroupoidCConvolutionAlgebra
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