Amenable group

Definition for locally compact groups

Equivalent conditions for amenability

• Existence of a left invariant mean on L^∞. The original definition, which depends on the axiom of choice.
• Existence of left-invariant states. There is a left-invariant state on any separable left-invariant unital C* subalgebra of the bounded continuous functions on G.
• Fixed-point property. Any action of the group by continuous affine transformations on a compact convex subset of a locally convex topological vector space has a fixed point. For locally compact abelian groups, this property is satisfied as a result of the Markov–Kakutani fixed-point theorem.
• Irreducible dual. All irreducible representations are weakly contained in the left regular representation λ on L².
• Trivial representation. The trivial representation of G is weakly contained in the left regular representation.
• Godement condition. Every bounded positive-definite measure μ on G satisfies μ ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ^–½f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.
• Day's asymptotic invariance condition. There is a sequence of integrable non-negative functions φ_n with integral 1 on G such that λφ_n − φ_n tends to 0 in the weak topology on L¹.
• Reiter's condition. For every finite subset F of G there is an integrable non-negative function φ with integral 1 such that λφ − φ is arbitrarily small in L¹ for g in F.
• Dixmier's condition. For every finite subset F of G there is unit vector f in L² such that λf − f is arbitrarily small in L² for g in F.
• Glicksberg−Reiter condition. For any f in L¹, the distance between 0 and the closed convex hull in L¹ of the left translates λf equals |∫f|.
• Følner condition. For every finite subset F of G there is a measurable subset U of G with finite positive Haar measure such that m/m is arbitrarily small for g in F.
• Leptin's condition. For every finite subset F of G there is a measurable subset U of G with finite positive Haar measure such that m/m is arbitrarily small.
• Kesten's condition. Left convolution on L² by a symmetric probability measure on G gives an operator of operator norm 1.
• Johnson's cohomological condition. The Banach algebra A = L¹ is amenable as a Banach algebra, i.e. any bounded derivation of A into the dual of a Banach A-bimodule is inner.

  Case of discrete groups

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.

• Γ is amenable.
• If Γ acts by isometries on a 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^∞ with μ = 1.
• There is a left invariant state μ on any left invariant separable unital C* subalgebra of ℓ^∞.
• 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.
• There are unit vectors x_n in ℓ² such that ||g · x_n − x_n||₂ tends to 0 for each g in Γ.
• 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.
• 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.
• If Γ acts by isometries on a Banach space E and f in ℓ^∞ is a bounded 1-cocycle, i.e. f = f + g·f, then f is a 1-coboundary, i.e. f = g·φ − φ for some φ in E*.
• The reduced group C*-algebra is nuclear.
• The reduced group C*-algebra is quasidiagonal.
• The von Neumann group algebra of Γ is hyperfinite.

Properties

• Every subgroup of an amenable group is amenable.
• Every quotient of an amenable group is amenable.
• A group extension of an amenable group by an amenable group is again amenable. In particular, finite direct product of amenable groups are amenable, although infinite products need not be.
• Direct limits of amenable groups are amenable. In particular, if a group can be written as a directed union of amenable subgroups, then it is amenable.
• Amenable groups are unitarizable; the converse is an open problem.
• Countable discrete amenable groups obey the Ornstein isomorphism theorem.

  Examples

• Finite groups are amenable. Use the counting measure with the discrete definition. More generally, compact groups are amenable. The Haar measure is an invariant mean.
• The group of integers is amenable. The existence of a shift-invariant, finitely additive probability measure on the group Z also follows easily from the Hahn–Banach theorem this way. Let S be the shift operator on the sequence space ℓ^∞, which is defined by _i = x_i+1 for all x ∈ ℓ^∞, and let u ∈ ℓ^∞ be the constant sequence u_i = 1 for all i ∈ Z. Any element y ∈ Y:=range has a distance larger than or equal to 1 from u. This implies that there is a well-defined norm-one linear form on the subspace Ru + Y taking tu + y to t. By the Hahn–Banach theorem the latter admits a norm-one linear extension on ℓ^∞, which is by construction a shift-invariant finitely additive probability measure on Z.
• If every conjugacy class in a locally compact group has compact closure, then the group is amenable. Examples of groups with this property include compact groups, locally compact abelian groups, and discrete groups with finite conjugacy classes.
• By the direct limit property above, a group is amenable if all its finitely generated subgroups are. That is, locally amenable groups are amenable.
• * By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable.
• It follows from the extension property above that a group is amenable if it has a finite index amenable subgroup. That is, virtually amenable groups are amenable.
• Furthermore, it follows that all solvable groups are amenable.

• Finitely generated groups of subexponential growth are amenable. A suitable subsequence of balls will provide a Følner sequence.
• Finitely generated infinite simple groups cannot be obtained by bootstrap constructions as used to construct elementary amenable groups. Since there exist such simple groups that are amenable, due to Juschenko and Monod, this provides again non-elementary amenable examples.

  Nonexamples