amenable group

Let $G$ be a locally compact group and $L^{\mathrm{\infty }}(G)$ be the Banach space of all essentially bounded functions $G\to ℝ$ with respect to the Haar measure.

Definition 1.

A linear functional on $L^{\mathrm{\infty }}(G)$ 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 $L_g$ be the left action of $g\in G$ on $f\in L^{\mathrm{\infty }}(G)$,i.e. $(L_{g}f)(h)=f(g^{-1}h)$.Then, a mean $\mu$ is said to be left invariant if $\mu (L_{g}f)=\mu (f)$for all $g\in G$ and $f\in L^{\mathrm{\infty }}(G)$.Similarly, right invariant if $\mu (R_{g}f)=\mu (f)$,where $R_g$ is the right action $(R_{g}f)(h)=f(hg)$.

Definition 3.

A locally compact group $G$ is amenable if there is a left (or right) invariant mean on $L^{\mathrm{\infty }}(G)$.

Example 1 (Amenable groups)

All finite groups and all abelian groups 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-abelian) subgroup on two generators then it is not amenable.