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class function (Definition)

Given a field $ K$, a $ K$-valued 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 $ \mathbb{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.



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Other names:  central function
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Cross-references: Haar measure, locally compact, central elements, complex, measurable, operation, convolution, property, convolution algebra, Lie group, compact, vector space, basis, irreducible, complex numbers, group representation, character, conjugacy class, function, group, field
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This is version 5 of class function, born on 2002-02-07, modified 2003-03-01.
Object id is 1847, canonical name is ClassFunction.
Accessed 3682 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
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