Unitary representations and modular actions
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Tome 326 (2005), pp. 97-144
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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.
@article{ZNSL_2005_326_a7,
author = {A. S. Kechris},
title = {Unitary representations and modular actions},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {97--144},
publisher = {mathdoc},
volume = {326},
year = {2005},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_326_a7/}
}
A. S. Kechris. Unitary representations and modular actions. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Tome 326 (2005), pp. 97-144. http://geodesic.mathdoc.fr/item/ZNSL_2005_326_a7/