Group structure and the pointwise ergodic theorem for connected amenable groups

Frederick P. Greenleaf, William R. Emerson

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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 {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.

Original languageEnglish (US)
Pages (from-to)153-172
Number of pages20
JournalAdvances in Mathematics
Volume14
Issue number2
DOIs
StatePublished - Oct 1974

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ASJC Scopus subject areas

  • Mathematics(all)

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