# amenable group

Let $G$ be a locally compact group and ${L}^{\mathrm{\infty}}(G)$ 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 ${L}^{\mathrm{\infty}}(G)$ 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 $f\in {L}^{\mathrm{\infty}}(G)$, 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 $f\in {L}^{\mathrm{\infty}}(G)$. 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 ${L}^{\mathrm{\infty}}(G)$.

###### 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.

Title | amenable group |
---|---|

Canonical name | AmenableGroup |

Date of creation | 2013-03-22 13:09:26 |

Last modified on | 2013-03-22 13:09:26 |

Owner | mhale (572) |

Last modified by | mhale (572) |

Numerical id | 9 |

Author | mhale (572) |

Entry type | Definition |

Classification | msc 43A07 |

Related topic | LpSpace |

Defines | amenable |

Defines | mean |