group C*-algebra

Let [G] be the group ring of a discrete group G. It has two completions to a C*-algebraPlanetmathPlanetmathPlanetmath:

Reduced group C*-algebra.

The reduced group C*-algebra, Cr*(G), is obtained by completing [G] in the operator normMathworldPlanetmath for its regular representation on l2(G).

Maximal group C*-algebra.

The maximal group C*-algebra, Cmax*(G) or just C*(G), is defined by the following universal propertyMathworldPlanetmath: any *-homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from [G] to some 𝔹() (the C*-algebra of bounded operatorsMathworldPlanetmath on some Hilbert spaceMathworldPlanetmath ) factors through the inclusion [G]Cmax*(G).

If G is amenable then Cr*(G)Cmax*(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