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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 9 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 17535 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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