ClassFunction 2002-02-07 13:31:33 2003-03-01 01:04:31 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\C}{\mathbb{C}} Given a field $K$, a $K$--valued {\em class function} on a group $G$ is a function $f: G \longrightarrow K$ 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. \paragraph{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.