class function

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

An important example of a class function is the character of a group representation. Over the complex numbers, the set of characters of the irreducible representations of $G$ form a basis for the vector space of all $ℂ$–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 $g\in C^{*}(G)$ (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.