### Abstract

Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group K = G R). We show how to explicitly construct sequences {U_{n}} of compacta in G in terms of the structural features of G which have the following property: For any "reasonable" action G × L^{p}(X, μ) ↓ L^{p}(X, μ) on an L^{p} space, 1 < p < ∞, and any f ∈ L^{p}(X, μ), the averages A_{n}f= 1 |U_{n}| ∫ U_{n}T_{g}
^{-1f}dg (|E|= left Haar measure inG). converge in L^{p} norm, and pointwise μ-a.e. on X, to G-invariant functions f* in L^{p}(X, μ). A single sequence {U_{n}} in G works for all L^{p} actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.

Original language | English (US) |
---|---|

Pages (from-to) | 153-172 |

Number of pages | 20 |

Journal | Advances in Mathematics |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - 1974 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*14*(2), 153-172. https://doi.org/10.1016/0001-8708(74)90027-9

**Group structure and the pointwise ergodic theorem for connected amenable groups.** / Greenleaf, Frederick P.; Emerson, William R.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 14, no. 2, pp. 153-172. https://doi.org/10.1016/0001-8708(74)90027-9

}

TY - JOUR

T1 - Group structure and the pointwise ergodic theorem for connected amenable groups

AU - Greenleaf, Frederick P.

AU - Emerson, William R.

PY - 1974

Y1 - 1974

N2 - Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group K = G R). We show how to explicitly construct sequences {Un} of compacta in G in terms of the structural features of G which have the following property: For any "reasonable" action G × Lp(X, μ) ↓ Lp(X, μ) on an Lp space, 1 < p < ∞, and any f ∈ Lp(X, μ), the averages Anf= 1 |Un| ∫ UnTg -1fdg (|E|= left Haar measure inG). converge in Lp norm, and pointwise μ-a.e. on X, to G-invariant functions f* in Lp(X, μ). A single sequence {Un} in G works for all Lp actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.

AB - Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group K = G R). We show how to explicitly construct sequences {Un} of compacta in G in terms of the structural features of G which have the following property: For any "reasonable" action G × Lp(X, μ) ↓ Lp(X, μ) on an Lp space, 1 < p < ∞, and any f ∈ Lp(X, μ), the averages Anf= 1 |Un| ∫ UnTg -1fdg (|E|= left Haar measure inG). converge in Lp norm, and pointwise μ-a.e. on X, to G-invariant functions f* in Lp(X, μ). A single sequence {Un} in G works for all Lp actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.

UR - http://www.scopus.com/inward/record.url?scp=19544390510&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=19544390510&partnerID=8YFLogxK

U2 - 10.1016/0001-8708(74)90027-9

DO - 10.1016/0001-8708(74)90027-9

M3 - Article

VL - 14

SP - 153

EP - 172

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -