class function

Given a field K, a K–valued class function on a group G is a function f:GK such that f(g)=f(h) whenever g and h are elements of the same conjugacy classMathworldPlanetmathPlanetmath of G.

An important example of a class function is the characterPlanetmathPlanetmath of a group representationMathworldPlanetmathPlanetmath. Over the complex numbersPlanetmathPlanetmath, the set of characters of the irreducible representations of G form a basis for the vector space of all –valued class functions, when G is a compact Lie group.

Relation to the convolution algebra

Class functions are also known as central functions, because they correspond to functions f in the convolution algebra C*(G) that have the property f*g=g*f for all gC*(G) (i.e., they commute with everything under the convolution operation). More precisely, the set of measurable complex valued class functions f is equal to the set of central elements of the convolution algebra C*(G), for G a locally compact group admitting a Haar measure.

Title class function
Canonical name ClassFunction
Date of creation 2013-03-22 12:18:06
Last modified on 2013-03-22 12:18:06
Owner djao (24)
Last modified by djao (24)
Numerical id 8
Author djao (24)
Entry type Definition
Classification msc 20A05
Synonym central function