![]() For every finite (or compact) subset F of G there is unit vector f in L 2( G) such that λ( g) f − f is arbitrarily small in L 2( G) for g in F. 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 L 1( G) for g in F. 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 L 1( G). Day's asymptotic invariance condition.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 –½ f has non-negative integral with respect to Haar measure, where Δ denotes the modular function. Every bounded positive-definite measure μ on G satisfies μ(1) ≥ 0. The trivial representation of G is weakly contained in the left regular representation. All irreducible representations are weakly contained in the left regular representation λ on L 2( G). For locally compact abelian groups, this property is satisfied as a result of the Markov–Kakutani fixed-point theorem. 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. There is a left-invariant state on any separable left-invariant unital C*-subalgebra of the bounded continuous functions on G. The original definition, which depends on the axiom of choice. ![]() Existence of a left (or right) invariant mean on L ∞( G).Pier (1984) contains a comprehensive account of the conditions on a second countable locally compact group G that are equivalent to amenability: A locally compact Hausdorff group is called amenable if it admits a left- (or right-)invariant mean.Įquivalent conditions for amenability ![]()
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