Amenable actions of locally compact groups

F. P. Greenleaf

Research output: Contribution to journalArticlepeer-review

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
DOIs
StatePublished - Oct 1969

ASJC Scopus subject areas

  • Analysis

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