On spectra of Koopman, groupoid and quasi-regular representations

Artem Dudko; Rostislav Grigorchuk

doi:10.3934/jmd.2017005

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2017, Volume 11: 99-123. Doi: 10.3934/jmd.2017005

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On spectra of Koopman, groupoid and quasi-regular representations

Artem Dudko^1, and
Rostislav Grigorchuk^2,

1.
Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, ON M5S 2E4, Canada
2.
Department of Mathematics, MS 3368, Texas A&M University, College Station, TX 77843-3368, USA

Author Bio: Artem Dudko artem.dudko@utoronto.ca; Rostislav Grigorchuk grigorch@math.tamu.edu

Received: May 30, 2016

Revised: November 04, 2016

Published: December 2016

Both authors were supported by the Swiss National Science Foundation.
RG:Supported by NSF grant DMS-1207699 and NSA grant H98230-15-1-0328.

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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.

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Mathematics Subject Classification: Primary: 20C15, 37A15.

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