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 • The measure is a
probability measure: the measure of the whole group
G is 1. • 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. • 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
μ on
G allows us to define integration of bounded functions on
G. Given a bounded function
f:
G →
R, the integral :\int_G f\,d\mu 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 \mu, it automatically has a bi-invariant one, which can be defined as :\nu(A) = \int_{g\in G}\!\!\mu (Ag) \, d\mu. 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
μn on Γ such that ||
g ·
μn −
μn||1 tends to 0 for each
g in Γ (M.M. Day). • There are unit vectors
xn in ℓ2(Γ) such that ||
g ·
xn −
xn||2 tends to 0 for each
g in Γ (J. Dixmier). • There are finite subsets
Sn of Γ such that |
g ·
Sn Δ
Sn| / |
Sn| 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
reduced group C*-algebra (see
the reduced group C*-algebra Cr*(G)) is
nuclear. • The
reduced group C*-algebra is quasidiagonal (J. Rosenberg, A. Tikuisis, S. White, W. Winter). • The
von Neumann group algebra (see
von Neumann algebras associated to groups) 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 connected groups. Amenability is related to
spectral theory of certain operators. For instance, the fundamental group of a closed Riemannian manifold is amenable if and only if the bottom of the spectrum of the
Laplacian on the
L2-space of the universal cover of the manifold is 0. ==Properties==