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 invariant 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.
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.
Read more about Amenable Group: Definition For Locally Compact Groups, Equivalent Conditions For Amenability, Case of Discrete Groups, Properties, Examples, Counterexamples
Famous quotes containing the words amenable and/or group:
“We must hold a man amenable to reason for the choice of his daily craft or profession. It is not an excuse any longer for his deeds that they are the custom of his trade. What business has he with an evil trade?”
—Ralph Waldo Emerson (1803–1882)
“[The Republicans] offer ... a detailed agenda for national renewal.... [On] reducing illegitimacy ... the state will use ... funds for programs to reduce out-of-wedlock pregnancies, to promote adoption, to establish and operate children’s group homes, to establish and operate residential group homes for unwed mothers, or for any purpose the state deems appropriate. None of the taxpayer funds may be used for abortion services or abortion counseling.”
—Newt Gingrich (b. 1943)