Amenable Group - Case of Discrete Groups

Case of Discrete Groups

The definition of amenability is simpler in the case of a discrete group, i.e. a group equipped with the discrete topology.

Definition. A discrete group G is amenable if there is a finitely additive measure (also called a mean) —a function that assigns to each subset of G a number from 0 to 1—such that

1. The measure is a probability measure: the measure of the whole group G is 1.
2. The measure is finitely additive: given finitely many disjoint subsets of G, the measure of the union of the sets is the sum of the measures.
3. The measure is left-invariant: given a subset A and an element g of G, the measure of A equals the measure of gA. (gA denotes the set of elements ga for each element a in A. That is, each element of A is translated on the left by g.)

This definition can be summarized thus: G is amenable if it has a finitely-additive left-invariant probability measure. Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?

It is a fact that this definition is equivalent to the definition in terms of L∞(G).

Having a measure on G allows us to define integration of bounded functions on G. Given a bounded function, the integral

is defined as in Lebesgue integration. (Note that some of the properties of the Lebesgue integral fail here, since our measure is only finitely additive.)

If a group has a left-invariant measure, it automatically has a bi-invariant one. Given a left-invariant measure, the function is a right-invariant measure. Combining these two gives a bi-invariant measure:

The equivalent conditions for amenability also become simpler in the case of a countable discrete group Γ. For such a group the following conditions are equivalent:

• Γ is amenable.
• If Γ acts by isometries on a (separable) Banach space E, leaving a weakly closed convex subset C of the closed unit ball of E* invariant, then Γ has a fixed point in C.
• There is a left invariant norm-continuous functional μ on l∞(Γ) with μ(1) = 1 (this requires the axiom of choice).
• There is a left invariant state μ on any left invariant separable unital C* subalgebra of l∞(Γ).
• There is a set of probability measures μ_n on Γ such that ||g · μ_n − μ_n||₁ tends to 0 for each g in Γ (M.M. Day).
• There are unit vectors x_n in l2(Γ) such that ||g · x_n − x_n||₂ tends to 0 for each g in Γ (J. Dixmier).
• There are finite subsets S_n of Γ such that | g · S_n Δ S_n | / |S_n| tends to 0 for each g in Γ (Følner).
• If μ is a symmetric probability measure on Γ with support generating Γ, then convolution by μ defines an operator of norm 1 on l2(Γ) (Kesten).
• If Γ acts by isometries on a (separable) Banach space E and f in l∞(Γ, E*) is a bounded 1-cocycle, i.e. f(gh) = f(g) + g·f(h), then f is a 1-coboundary, i.e. f(g) = g·φ − φ for some φ in E* (B.E. Johnson).
• The von Neumann group algebra of Γ is hyperfinite (A. Connes).

Note that A. Connes also proved that the von Neumann group algebra of any connected locally compact group is hyperfinite, so the last condition no longer applies in the case of locally compact groups.