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

单词	GroupCalgebra
释义	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 $C^{}_{r}(G)\\cong C^{}_{\\mathrm{max}}(G)$ .
随便看	convergence condition of infinite product ConvergenceCriterionForInfiniteProduct convergence criterion for infinite product ConvergencedivergenceForAnInfiniteProduct convergence/divergence for an infinite product ConvergenceInDistribution convergence in distribution ConvergenceInProbability convergence in probability ConvergenceInProbabilityIsPreservedUnderContinuousTransformations convergence in probability is preserved under continuous transformations ConvergenceInTheMean convergence in the mean ConvergenceOfArithmeticgeometricMean convergence of arithmetic-geometric mean ConvergenceOfASequenceWithFiniteUpcrossings convergence of a sequence with finite upcrossings ConvergenceOfComplexTermSeries convergence of complex term series ConvergenceOfIntegrals convergence of integrals ConvergenceOfRiemannZetaSeries convergence of Riemann zeta series ConvergenceOfTheSequence11nn convergence of the sequence (1+1/n)^n

group C*-algebra

group $C^{*}$ -algebra