The University of Southampton
University of Southampton Institutional Repository

Amenable actions, invariant means and bounded cohomology

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.
1793-5253
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
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, 321-334. (doi:10.1142/S1793525312500161).

Record type: Article

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
Download (167kB)

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: https://eprints.soton.ac.uk/id/eprint/143057
ISSN: 1793-5253
PURE UUID: abdc743d-2198-4bb4-b00f-3eae9b3ada6b

Catalogue record

Date deposited: 08 Apr 2010 09:18
Last modified: 13 Mar 2019 20:10

Export record

Altmetrics

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of https://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×