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'amenable group'
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| Title of object: |
amenable group |
| Canonical Name: |
AmenableGroup |
| Type: |
Definition |
| Created on: |
2002-11-15 00:16:19.390047-05 |
| Modified on: |
2002-11-15 00:16:19.390047-05 |
| Classification: |
msc:43A07 |
| Defines: |
amenable, mean |
Revision comment (for changes between this and next version):
| Changes for correction #1247 ('clarification'). |
Preamble:
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Content:
Let $G$ be a locally compact group and $L^\infty(G)$ be the Banach space of all bounded functions $G \to \Rset$ with respect to the Haar measure.
\begin{definition}
A linear functional on $L^\infty(G)$ is called a \defn{mean} if it maps the constant function $f(g) = 1$ to 1 and non-negative functions to non-negative numbers.
\end{definition}
\begin{definition}
Let $L_g$ be the left action of $g \in G$ on $f \in L^\infty(G)$,
i.e.\ $(L_g f)(h) = f(gh)$.
Then, a mean $\mu$ is said to be \defn{left invariant} if $\mu(L_g f) = \mu(f)$
for all $g \in G$ and $f \in L^\infty(G)$.
Similarly, \defn{right invariant} if $\mu(R_g f) = \mu(f)$,
where $R_g$ is the right action $(R_g f)(h) = f(hg)$.
\end{definition}
\begin{definition}
A locally compact group $G$ is \defn{amenable} if there is a left (or right) invariant mean on $L^\infty(G)$.
\end{definition}
All finite groups and all abelian groups are amenable.
If a group contains a free subgroup on two generators then it is not amenable. |
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