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G - Amenability

Published online by Cambridge University Press:  12 September 2009

Bachir Bekka
Affiliation:
Université de Rennes I, France
Pierre de la Harpe
Affiliation:
Université de Genève
Alain Valette
Affiliation:
Université de Neuchatel, Switzerland
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Summary

Lebesgue proposed his view on integration in a short note (1901) and a famous book (1904). It was then a natural question to know whether Lebesgue's measure could be extended as a finitely additive measure defined on all subsets of Rn which is invariant under isometric transformations. Hausdorff answered negatively for n ≥ 3 in 1914 and Banach positively for n ≤ 2 in 1923. Von Neumann showed in 1929 that the deep reason for this difference lies in the group of isometries of Rn (viewed as a discrete group) which is amenable for n ≤ 2 and which is not so for n ≥ 3. In 1950, Dixmier extended the notion of amenability to topological groups [Dixmi–50].

In Section G.1, amenability is defined for a topological group in two equivalent ways, by the existence of an invariant mean on an appropriate space and as a fixed point property. Examples are given in Section G.2. For a locally compact group G, it is shown in Section G.3 that G is amenable if and only if 1GλG; a consequence is that amenability is inherited by closed subgroups of locally compact groups.

It is remarkable that amenability can be given a very large number of equivalent definitions, at least for locally compact groups. In Section G.4, we give Kesten's characterisation in terms of appropriate operators, which is crucial for the study of random walks on groups.

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Kazhdan's Property (T) , pp. 420 - 448
Publisher: Cambridge University Press
Print publication year: 2008

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  • Amenability
  • Bachir Bekka, Université de Rennes I, France, Pierre de la Harpe, Université de Genève, Alain Valette, Université de Neuchatel, Switzerland
  • Book: Kazhdan's Property (T)
  • Online publication: 12 September 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542749.015
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  • Amenability
  • Bachir Bekka, Université de Rennes I, France, Pierre de la Harpe, Université de Genève, Alain Valette, Université de Neuchatel, Switzerland
  • Book: Kazhdan's Property (T)
  • Online publication: 12 September 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542749.015
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Amenability
  • Bachir Bekka, Université de Rennes I, France, Pierre de la Harpe, Université de Genève, Alain Valette, Université de Neuchatel, Switzerland
  • Book: Kazhdan's Property (T)
  • Online publication: 12 September 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542749.015
Available formats
×