### 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) |
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Pages (from-to) | 153-172 |

Number of pages | 20 |

Journal | Advances in Mathematics |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Oct 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