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 $C_r^{*}(G)\cong C_{\max }^{*}(G)$.

Title	group $C^{*}$-algebra
Canonical name	GroupCalgebra
Date of creation	2013-03-22 13:10:53
Last modified on	2013-03-22 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