Journal of Modern Dynamics (JMD)

On spectra of Koopman, groupoid and quasi-regular representations

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

       Abstract        References        Full Text (282.7K)       Related Articles       

Artem Dudko - Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, ON M5S 2E4, Canada (email)
Rostislav Grigorchuk - Department of Mathematics, MS 3368, Texas A&M University, College Station, TX 77843-3368, United States (email)

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:  Koopman representation, groupoid construction, quasi-regular representation, spectrum, weak containment, weakly branch groups.
Mathematics Subject Classification:  Primary: 20C15, 37A15.

Received: May 2016;      Revised: November 2016;      Available Online: January 2017.