group ${C}^{*}$algebra
Let $\u2102[G]$ 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 $\u2102[G]$ 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 $\u2102[G]$ to some $\mathbb{B}(\mathscr{H})$ (the ${C}^{*}$algebra of bounded operators^{} on some Hilbert space^{} $\mathscr{H}$) factors through the inclusion $\u2102[G]\hookrightarrow {C}_{\mathrm{max}}^{*}(G)$.
If $G$ is amenable then ${C}_{r}^{*}(G)\cong {C}_{\mathrm{max}}^{*}(G)$.
Title  group ${C}^{*}$algebra 

Canonical name  GroupCalgebra 
Date of creation  20130322 13:10:53 
Last modified on  20130322 13:10:53 
Owner  mhale (572) 
Last modified by  mhale (572) 
Numerical id  6 
Author  mhale (572) 
Entry type  Definition 
Classification  msc 22D15 
Related topic  CAlgebra 
Related topic  GroupoidCConvolutionAlgebra 