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amenable group
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(Definition)
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Let $G$ be a locally compact group and $L^\infty(G)$ be the Banach space of all essentially bounded functions $G \to \Rset$ with respect to the Haar measure.
Definition 2 Let $L_g$ be the left action of $g \in G$ on $f \in L^\infty(G)$ , i.e. $(L_g f)(h) = f(g^{-1}h)$ . Then, a mean $\mu$ is said to be left invariant if $\mu(L_g f) = \mu(f)$ for all $g \in G$ and $f \in L^\infty(G)$ . Similarly, right invariant if $\mu(R_g f) = \mu(f)$ , where $R_g$ is the right action $(R_g f)(h) = f(hg)$ .
Definition 3 A locally compact group $G$ is amenable if there is a left (or right) invariant mean on $L^\infty(G)$ .
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"amenable group" is owned by mhale.
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See Also:  -space
| Also defines: |
amenable, mean |
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Cross-references: generators, subgroup, non-Abelian, contains, compact, abelian groups, finite groups, right action, right, invariant, left action, numbers, functions, constant function, maps, linear functional, Haar measure, essentially bounded functions, Banach space, group, locally compact
There are 14 references to this entry.
This is version 6 of amenable group, born on 2002-11-15, modified 2005-02-02.
Object id is 3598, canonical name is AmenableGroup.
Accessed 18229 times total.
Classification:
| AMS MSC: | 43A07 (Abstract harmonic analysis :: Means on groups, semigroups, etc.; amenable groups) |
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Pending Errata and Addenda
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