Locally compact topological group with an invariant averaging operation
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".[a]
The critical step in the Banach–Tarski paradox construction is to find inside the rotation group SO(3) a free subgroup on two generators. Amenable groups cannot contain such groups, and do not allow this kind of paradoxical construction.
Amenability has many equivalent definitions. 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. For example, any subgroup of the group of integers is generated by some integer . If then the subgroup takes up 0 proportion. Otherwise, it takes up of the whole group. Even though both the group and the subgroup has infinitely many elements, there is a well-defined sense of proportion.
If a group has a Følner sequence then it is automatically amenable.
^Day 1949, pp. 1054–1055.
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In mathematics, an amenablegroup is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant...
Look up amenable in Wiktionary, the free dictionary. Amenable may refer to: AmenablegroupAmenable species Amenable number Amenable set Agreeableness...
Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture...
sofic groups. The limit of a sequence of amenablegroups (that is, an initially subamenable group) is necessarily sofic, but there exist sofic groups that...
additional assumptions) from finite subsets of an amenablegroup , and further, of a cancellative left-amenable semigroup. Theorem: — For every measurable subadditive...
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Tarski Monster groups for each prime p > 10 75 {\displaystyle p>10^{75}} . Tarski monster groups are examples of non-amenablegroups not containing any...
asymptotic cones of finitely generated groups (see e.g.); amenability of a finitely generated group; being virtually abelian (that is, having an abelian subgroup...
not contain a subgroup isomorphic to the free group of rank 2. It is conjectured that F is not amenable and hence a further counterexample to the long-standing...
Prigozhin's had spotted an opening, he would pitch it to Mr. Putin, and, if amenable, Mr. Putin would unofficially sanction Wagner's operations, sometimes providing...
A is amenable if and only if it has a virtual diagonal. If A is a group algebra L 1 ( G ) {\displaystyle L^{1}(G)} for some locally compact group G then...
particular free groups) Elementary amenablegroups (e.g. virtually abelian groups) Diffuse groups – in particular, groups that act freely isometrically on...
the Ornstein isomorphism theorem still holds if the group G is a countably infinite amenablegroup. An invertible, measure-preserving transformation of...
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many groups with these properties, as any locally finite group is amenable. For example, the von Neumann group algebra of the infinite symmetric group of...
locally compact amenablegroups, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenablegroups such as SL(2,R)...
{\displaystyle s} . Grigorchuk's group was also the first example of a group that is amenable but not elementary amenable, thus answering a problem posed...
equivalence relation is hyperfinite on the remaining space. Any group which is not amenable admits a Borel action on a standard Borel space which induces...
of actions of semisimple groups, and introduced the basic notion of amenablegroup action. Zimmer was married to Terese Schwartzman, former director of...
C*(G). The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if G is amenable. The group von Neumann algebra...
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