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 $C^{*}_{r}(G)\cong C^{*}_{\mathrm{max}}(G)$.

Title group $C^{*}$-algebra GroupCalgebra 2013-03-22 13:10:53 2013-03-22 13:10:53 mhale (572) mhale (572) 6 mhale (572) Definition msc 22D15 CAlgebra GroupoidCConvolutionAlgebra