# class function

Given a field $K$, a $K$–valued class function on a group $G$ is a function $f:G\u27f6K$ 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 $\u2102$–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.

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 |