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amenable group (Definition)

Let $ G$ be a locally compact group and $ L^\infty(G)$ be the Banach space of all essentially bounded functions $ G \to \mathbb{R}$ with respect to the Haar measure.

Definition 1   A linear functional on $ L^\infty(G)$ is called a mean if it maps the constant function $ f(g) = 1$ to 1 and non-negative functions to non-negative numbers.
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)$.
Example 1 (Amenable groups)
All finite groups and all abelian groups are amenable. Compact groups are amenable as the Haar measure is an (unique) invariant mean.
Example 2 (Non-amenable groups)
If a group contains a free (non-abelian) subgroup on two generators then it is not amenable.



"amenable group" is owned by mhale.
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See Also: $L^p$-space

Also defines:  amenable, mean
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Cross-references: generators, subgroup, contains, compact, abelian groups, finite groups, right action, right, invariant, left action, functions, constant function, maps, linear functional, Haar measure, essentially bounded functions, Banach space, group, locally compact
There are 23 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 15835 times total.

Classification:
AMS MSC: 43A07 (Abstract harmonic analysis :: Means on groups, semigroups, etc.; amenable groups)
Pending Errata and Addenda
None.
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