12/12/2023 0 Comments Invariant subspace definition![]() 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 f) = Λ( f) for all g in G, and f in L ∞( G) with respect to the left (respectively right) shift action of gĭefinition 3.A mean Λ in Hom( L ∞( G), R) is said to be left-invariant (respectively right-invariant) if Λ( g 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. (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).ĭefinition 1. Then it is well known that it possesses a unique, up-to-scale left- (or right-) translation invariant nontrivial ring measure, the Haar measure. Let G be a locally compact Hausdorff group. If a group has a Følner sequence then it is automatically amenable.ĭefinition for locally compact groups In this setting, a group is amenable if one can say what proportion of G any given subset takes up. In discrete group theory, where G has the discrete topology, a simpler definition is used. An intuitive way to understand this version is that the support of the regular representation is the whole space of irreducible representations. In the field of analysis, the definition is in terms of linear functionals. The amenability property has a large number of equivalent formulations. Day introduced the English translation "amenable", apparently as a pun on " mean". The original definition, in terms of a finitely additive measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. Locally compact topological group with an invariant averaging operation
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