Amenable actions, invariant means and bounded cohomology
Amenable actions, invariant means and bounded cohomology
We show that topological amenability of an action of a countable discrete group on a compact space
is equivalent to the existence of an invariant
mean for the action. We prove also that this is equivalent to vanishing of
bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class.
In the case when the compact space is a point our result reduces to a classic theorem
of B.E.~Johnson characterising amenability of groups. In the case when the compact
space is the Stone-\v{C}ech compactification of the group we obtain a cohomological characterisation
of exactness for the group, answering a question of Higson.
321-334
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944
Nowak, Piotr
d86850e8-5309-4f78-abf0-73d2036249b1
Wright, Nick
f4685b8d-7496-47dc-95f0-aba3f70fbccd
14 August 2012
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944
Nowak, Piotr
d86850e8-5309-4f78-abf0-73d2036249b1
Wright, Nick
f4685b8d-7496-47dc-95f0-aba3f70fbccd
Brodzki, Jacek, Niblo, Graham A., Nowak, Piotr and Wright, Nick
(2012)
Amenable actions, invariant means and bounded cohomology.
Journal of Topology and Analysis, 4 (321), Autumn Issue, .
(doi:10.1142/S1793525312500161).
Abstract
We show that topological amenability of an action of a countable discrete group on a compact space
is equivalent to the existence of an invariant
mean for the action. We prove also that this is equivalent to vanishing of
bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class.
In the case when the compact space is a point our result reduces to a classic theorem
of B.E.~Johnson characterising amenability of groups. In the case when the compact
space is the Stone-\v{C}ech compactification of the group we obtain a cohomological characterisation
of exactness for the group, answering a question of Higson.
Text
Amenable_actions_Final_Draft.pdf
- Author's Original
More information
Submitted date: 2 September 2010
Accepted/In Press date: 8 July 2012
Published date: 14 August 2012
Additional Information:
Funded by National Science Foundation: Isoperimetric inequalities and the large-scale geometry of groups (900874)
Organisations:
Pure Mathematics
Identifiers
Local EPrints ID: 143057
URI: http://eprints.soton.ac.uk/id/eprint/143057
ISSN: 1793-5253
PURE UUID: abdc743d-2198-4bb4-b00f-3eae9b3ada6b
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Date deposited: 08 Apr 2010 09:18
Last modified: 16 Mar 2024 03:43
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Author:
Piotr Nowak
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