# amenable group

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

###### Definition 1.

A linear functional on $L^{\infty}(G)$ 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 $f\in L^{\infty}(G)$, 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 $f\in L^{\infty}(G)$. 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 $L^{\infty}(G)$.

###### 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.

Title amenable group AmenableGroup 2013-03-22 13:09:26 2013-03-22 13:09:26 mhale (572) mhale (572) 9 mhale (572) Definition msc 43A07 LpSpace amenable mean