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Unitary representations and modular actions

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Abstract

We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e., in a precise sense “orthogonal” to any Borel action of a countable group by automorphisms on a countable rooted tree. We also study tempered actions of countable groups by automorphisms on compact metrizable groups, where it turns out that this notion has several ergodic theoretic reformulations and fits naturally in a hierarchy of strong ergodicity properties strictly between ergodicity and strong mixing. Bibliography:s 25 titles.

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Authors and Affiliations

  1. Department of Mathematics, California Institute of Technology, Pasadena, California, USA

    A. S. Kechris

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  1. A. S. Kechris
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Dedicated to Professor Anatoly Vershik on the occasion of his 70th birthday

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 97–144.

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Kechris, A.S. Unitary representations and modular actions. J Math Sci 140, 398–425 (2007). https://doi.org/10.1007/s10958-007-0449-y

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  • DOI: https://doi.org/10.1007/s10958-007-0449-y

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