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amenable group
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 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.
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.
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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