Amenable group

Let G be a locally compact Hausdorff group. Then it is well known that it possesses a unique, up-to-scale left- (or right-) translation invariant nontrivial ring measure, the Haar measure. (This is a Borel regular measure when G is second-countable; there are both left and right measures when G is compact.) Consider the Banach space L^∞(G) of essentially bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).

Definition 1. A linear functional Λ in Hom(L^∞(G), R) is said to be a mean if Λ has norm 1 and is non-negative, i.e. f ≥ 0 a.e. implies Λ(f) ≥ 0.

Definition 2. A mean Λ in Hom(L^∞(G), R) is said to be left-invariant (respectively right-invariant) if Λ(g·f) = Λ(f) for all g in G, and f in L^∞(G) with respect to the left (respectively right) shift action of g·f(x) = f(g⁻¹x) (respectively f·g(x) = f(xg⁻¹)).

Definition 3. A locally compact Hausdorff group is called amenable if it admits a left- (or right-)invariant mean.

By identifying Hom(L^∞(G), R) with the space of finitely-additive Borel measures which are absolutely continuous with respect to the Haar measure on G (a ba space), the terminology becomes more natural: a mean in Hom(L^∞(G), R) induces a left-invariant, finitely additive Borel measure on G which gives the whole group weight 1.

As an example for compact groups, consider the circle group. The graph of a typical function f ≥ 0 looks like a jagged curve above a circle, which can be made by tearing off the end of a paper tube. The linear functional would then average the curve by snipping off some paper from one place and gluing it to another place, creating a flat top again. This is the invariant mean, i.e. the average value ${\displaystyle \Lambda (f)=\int _{\mathbb {R} /\mathbb {Z} }f\ d\lambda }$  where ${\displaystyle \lambda }$  is Lebesgue measure.

Left-invariance would mean that rotating the tube does not change the height of the flat top at the end. That is, only the shape of the tube matters. Combined with linearity, positivity, and norm-1, this is sufficient to prove that the invariant mean we have constructed is unique.

As an example for locally compact groups, consider the group of integers. A bounded function f is simply a bounded function of type ${\displaystyle f:\mathbb {Z} \to \mathbb {R} }$ , and its mean is the running average ${\displaystyle \lim _{n}{\frac {1}{2n+1}}\sum _{k=-n}^{n}f(k)}$ .

Pier (1984) contains a comprehensive account of the conditions on a second countable locally compact group G that are equivalent to amenability:^[2]

• Existence of a left (or right) invariant mean on L^∞(G). 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 (separable) 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(G).
• 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(1) ≥ 0. Valette 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^–1⁄2f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.^[3]
• Day's asymptotic invariance condition. There is a sequence of integrable non-negative functions φ_n with integral 1 on G such that λ(g)φ_n − φ_n tends to 0 in the weak topology on L¹(G).
• Reiter's condition. For every finite (or compact) subset F of G there is an integrable non-negative function φ with integral 1 such that λ(g)φ − φ is arbitrarily small in L¹(G) for g in F.
• Dixmier's condition. For every finite (or compact) subset F of G there is unit vector f in L²(G) such that λ(g)f − f is arbitrarily small in L²(G) for g in F.
• Glicksberg−Reiter condition. For any f in L¹(G), the distance between 0 and the closed convex hull in L¹(G) of the left translates λ(g)f equals |∫f|.
• Følner condition. For every finite (or compact) subset F of G there is a measurable subset U of G with finite positive Haar measure such that m(U Δ gU)/m(U) is arbitrarily small for g in F.
• Leptin's condition. For every finite (or compact) subset F of G there is a measurable subset U of G with finite positive Haar measure such that m(FU Δ U)/m(U) is arbitrarily small.
• Kesten's condition. Left convolution on L²(G) by a symmetric probability measure on G gives an operator of operator norm 1.
• Johnson's cohomological condition. The Banach algebra A = L¹(G) is amenable as a Banach algebra, i.e. any bounded derivation of A into the dual of a Banach A-bimodule is inner.

The definition of amenability is simpler in the case of a discrete group,^[4] i.e. a group equipped with the discrete topology.^[5]

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 μ on G allows us to define integration of bounded functions on G. Given a bounded function f: G → R, the integral

${\displaystyle \int _{G}f\,d\mu }$ 

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, it automatically has a bi-invariant one. Given a left-invariant measure μ, the function μ⁻(A) = μ(A⁻¹) is a right-invariant measure. Combining these two gives a bi-invariant measure:

${\displaystyle \nu (A)=\int _{g\in G}\mu \left(Ag^{-1}\right)\,d\mu ^{-}.}$ 

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:^[2]

• 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 reduced group C*-algebra (see the reduced group C*-algebra C_r*(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.^[6]

• Every (closed) 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.^[7][8]

• 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 (unique taking total measure 1).
• The group of integers is amenable (a sequence of intervals of length tending to infinity is a Følner sequence). 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^∞(Z), which is defined by (Sx)_i = x_i+1 for all x ∈ ℓ^∞(Z), and let u ∈ ℓ^∞(Z) be the constant sequence u_i = 1 for all i ∈ Z. Any element y ∈ Y:=range(S − I) has a distance larger than or equal to 1 from u (otherwise y_i = x_i+1 - x_i would be positive and bounded away from zero, whence x_i could not be bounded). 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^∞(Z), 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.^[9]
• 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.

All examples above are elementary amenable. The first class of examples below can be used to exhibit non-elementary amenable examples thanks to the existence of groups of intermediate growth.

• Finitely generated groups of subexponential growth are amenable. A suitable subsequence of balls will provide a Følner sequence.^[10]
• 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,^[11] this provides again non-elementary amenable examples.

If a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable. The converse to this statement is the so-called von Neumann conjecture, which was disproved by Olshanskii in 1980 using his Tarski monsters. Adyan subsequently showed that free Burnside groups are non-amenable: since they are periodic, they cannot contain the free group on two generators. These groups are finitely generated, but not finitely presented. However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely presented groups that have a periodic normal subgroup with quotient the integers.^[12]

For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative:^[13] every subgroup of GL(n,k) with k a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators. Although Tits' proof used algebraic geometry, Guivarc'h later found an analytic proof based on V. Oseledets' multiplicative ergodic theorem.^[14] Analogues of the Tits alternative have been proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes of non-positive curvature.^[15]

• Uniformly bounded representation
• Kazhdan's property (T)
• Von Neumann conjecture

This article incorporates material from Amenable group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

• Some notes on amenability by Terry Tao
• Garrido, Alejandra. An introduction to amenable groups