### 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 L^{1}(Z, ν), and this induces a natural adjoint action on L^{∞}(Z, ν) = L^{1}(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 language | English (US) |
---|---|

Pages (from-to) | 295-315 |

Number of pages | 21 |

Journal | Journal of Functional Analysis |

Volume | 4 |

Issue number | 2 |

State | Published - Oct 1969 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*4*(2), 295-315.

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

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 4, no. 2, pp. 295-315.

}

TY - JOUR

T1 - Amenable actions of locally compact groups

AU - Greenleaf, F. P.

PY - 1969/10

Y1 - 1969/10

N2 - 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.

AB - 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.

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M3 - Article

VL - 4

SP - 295

EP - 315

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -