amenable group

Let G be a locally compact group and L(G) be the Banach spaceMathworldPlanetmath of all essentially bounded functions G with respect to the Haar measure.

Definition 1.

A linear functionalMathworldPlanetmath on L(G) is called a mean if it maps the constant function f(g)=1 to 1 and non-negative functions to non-negative numbers.

Definition 2.

Let Lg be the left action of gG on fL(G), i.e. (Lgf)(h)=f(g-1h). Then, a mean μ is said to be left invariant if μ(Lgf)=μ(f) for all gG and fL(G). Similarly, right invariant if μ(Rgf)=μ(f), where Rg is the right action (Rgf)(h)=f(hg).

Definition 3.

A locally compact group G is amenable if there is a left (or right) invariant mean on L(G).

Example 1 (Amenable groups)

All finite groupsMathworldPlanetmath and all abelian groupsMathworldPlanetmath are amenable. Compact groups are amenable as the Haar measure is an (unique) invariant mean.

Example 2 (Non-amenable groups)

If a group contains a free (non-abelianMathworldPlanetmath) subgroupMathworldPlanetmathPlanetmath on two generatorsPlanetmathPlanetmathPlanetmath then it is not amenable.

Title amenable group
Canonical name AmenableGroup
Date of creation 2013-03-22 13:09:26
Last modified on 2013-03-22 13:09:26
Owner mhale (572)
Last modified by mhale (572)
Numerical id 9
Author mhale (572)
Entry type Definition
Classification msc 43A07
Related topic LpSpace
Defines amenable
Defines mean