Amenable group: Difference between revisions

Latest revision as of 10:42, 25 July 2008

Definition

An amenable group is a locally compact topological group that can be equipped with an additional structure of a left (or right) invariant mean. A mean on a locally compact group $G$ is a linear functional on $L^{\infty} (G)$ (the Banach space of essentially bounded functions from $G$ to $R$ ) that maps nonnegative functions to nonnegative functions and sends the constant function (valuing everything to 1) to 1.

By left-invariant we mean that the mean is invariant under the action of the group on the space $L^{\infty} (G)$ .

We can also define amenability purely in the context of discrete groups, in which case the definition becomes far simpler. Check out amenable discrete group.

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 ==Definition==
-An '''amenable group'''' is a [[locally compact group|locally compact]] [[topological group]] equipped with an additional structure of a left (or right) invariant '''mean'''. A mean on a locally compact group <math>G</math> is a linear functional on <math>L^\infty(G)</math> (the Banach space of essentially bounded functions from <math>G</math> to <math>\R</math>) that maps nonnegative functions to nonnegative functions and sends the constant function (valuing everything to 1) to 1.
+An '''amenable group''' is a [[locally compact group|locally compact]] [[topological group]] that can be equipped with an additional structure of a left (or right) invariant '''mean'''. A mean on a locally compact group <math>G</math> is a linear functional on <math>L^\infty(G)</math> (the Banach space of essentially bounded functions from <math>G</math> to <math>\R</math>) that maps nonnegative functions to nonnegative functions and sends the constant function (valuing everything to 1) to 1.
 By '''left-invariant''' we mean that the mean is invariant under the action of the group on the space <math>L^\infty(G)</math>.
 We can also define amenability purely in the context of discrete groups, in which case the definition becomes far simpler. Check out [[amenable discrete group]].