Amenable Group - Equivalent Conditions For Amenability

Equivalent Conditions For Amenability

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

• 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 (1998) 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 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 L1(G) by a 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.