Amenable Groups

Throughout we let Γ be a discrete group. For f : Γ → C and each s ∈ Γ we define the left translation action by (s.f)(t) = f(s−1t). Definition 1.1. A group Γ is amenable is there exists a state μ on l∞(Γ) which is invariant under the left translation action: for all s ∈ Γ and f ∈ l∞(Γ), μ(s.f) = μ(f). Example 1.2. Finite groups are amenable: take the state which sends χ{s} to 1 |Γ| for each s ∈ Γ. One can also see that abelian groups are amenable through the Markov-Kakutani fixed point theorem. Furthermore, the class of amenable groups is closed under taking subgroups, extensions, quotients, and inductive limits. Hence we can construct further examples from finite and abelian groups.