On spectra of Koopman, groupoid and quasi-regular representations
Abstract
In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corollary of our results we obtain a continuum of pairwise disjoint pairwise equivalent irreducible representations of weakly branch groups. As an illustration we calculate spectra of regular, Koopman and groupoid representations associated to the action of the 2-group of intermediate growth constructed by the second author in 1980.
Keywords
Cite
@article{arxiv.1510.00897,
title = {On spectra of Koopman, groupoid and quasi-regular representations},
author = {Artem Dudko and Rostislav Grigorchuk},
journal= {arXiv preprint arXiv:1510.00897},
year = {2017}
}
Comments
Some typos are fixed (including in Theorem 1 and in the proof of Kuhn's Theorem). Several passages not directly relevant to the proofs are removed. Other minor changes are performed