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Published January 2007 | public
Journal Article

Unitary representations and modular actions

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.

Additional Information

© 2007 Springer. Research partially supported by NSF Grant DMS-9987437 and a Guggenheim Fellowship. Dedicated to Professor Anatoly Vershik on the occasion of his 70th birthday.

Additional details

Created:
August 19, 2023
Modified:
March 5, 2024