amenable group
Let be a locally compact group and be the Banach space of all essentially bounded functions with respect to the Haar measure.
Definition 1.
A linear functional on is called a mean if it maps the constant function to 1 and non-negative functions to non-negative numbers.
Definition 2.
Let be the left action of on , i.e. . Then, a mean is said to be left invariant if for all and . Similarly, right invariant if , where is the right action .
Definition 3.
A locally compact group is amenable if there is a left (or right) invariant mean on .
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 |