Amenable group

Definition

An amenable group' is a locally compact topological group equipped with an additional structure of a left (or right) invariant mean. A mean on a locally compact group $G$ is a linear functional on $L^{\mathrm{\infty }}(G)$ (the Banach space of essentially bounded functions from $G$ to $\mathbb{R}$) that maps nonnegative functions to nonnegative functions and sends the constant function (valuing everything to 1) to 1.

By left-invariant we mean that the mean is invariant under the action of the group on the space $L^{\mathrm{\infty }}(G)$.

We can also define amenability purely in the context of discrete groups, in which case the definition becomes far simpler. Check out amenable discrete group.