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Concept
Amenable group
Summary
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. 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 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".
The amenability property has a large number of equivalent formulations. In the field of analysis, the definition is in terms of linear functionals. An intuitive way to understand this version is that the support of the regular representation is the whole space of irreducible representations.
In discrete group theory, where G has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one can say what proportion of G any given subset takes up.
If a group has a Følner sequence then it is automatically amenable.
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.
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