Amenable actions of locally compact groups

F. P. Greenleaf

Research output: Contribution to journalArticle

Abstract

We consider a locally compact group G with jointly continuous action G × Z → Z on a locally compact space. The finite Radon (regular Borel) measures M(G) act naturally on various function spaces supported on Z, including the continuous bounded functions CB(Z). If Z supports a (not necessarily unique) nonnegative quasi-invariant measure ν we apply some recent studies [7] to define a Banach module action of M(G) on L1(Z, ν), and this induces a natural adjoint action on L(Z, ν) = L1(Z, ν)*. These actions of M(G) (and of G) give us definitions of amenability of the action of G on various function spaces supported on Z, including CB(Z) and L(Z, ν) (when there is a quasi-invariant ν on Z), corresponding to the existence of a left-invariant mean on these various spaces. When G acts on one of its coset spaces solG H, H a closed subgroup, there is always a quasi-invariant measure on G H and the different definitions of amenability of the action G × G H → G H all coincide. A number of manipulations of invariant means on groups (the case where G = Z) then carry over to the context of transformation groups. We apply them to explore the following problems. Let G × G H → G H be an amenable action of G on one of its coset spaces. Let ν be a quasi-invariant measure on G H.

Original languageEnglish (US)
Pages (from-to)295-315
Number of pages21
JournalJournal of Functional Analysis
Volume4
Issue number2
StatePublished - Oct 1969

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Locally Compact Group
Invariant Measure
Invariant Mean
Amenability
Coset
Function Space
Banach Module
Locally Compact Space
Transformation group
H-space
Borel Measure
Manipulation
Non-negative
Subgroup
Closed
Invariant

ASJC Scopus subject areas

  • Analysis

Cite this

Amenable actions of locally compact groups. / Greenleaf, F. P.

In: Journal of Functional Analysis, Vol. 4, No. 2, 10.1969, p. 295-315.

Research output: Contribution to journalArticle

Greenleaf, FP 1969, 'Amenable actions of locally compact groups', Journal of Functional Analysis, vol. 4, no. 2, pp. 295-315.
Greenleaf, F. P. / Amenable actions of locally compact groups. In: Journal of Functional Analysis. 1969 ; Vol. 4, No. 2. pp. 295-315.
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