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On spectra of Koopman, groupoid and quasi-regular representations

Amenability, coarse embeddability and fixed point properties December 06, 2016 - December 09, 2016

December 09, 2016 (02:00 PM PST - 03:00 PM PST)
Speaker(s): Rostislav Grigorchuk (Texas A & M University)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • Unitary representations

  • koopman representation

  • groupoid representation

  • spectra of representations

  • groups of intermediate growth

  • Amenability

  • a-T-menability

  • fixed point properties

  • hyperbolic groups and generalizations

  • Banach space

  • group cohomology

  • expander graph

  • index theory

  • non-commutative geometry

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Abstract

Given an action of a countable group on a probability measure space by a measure class preserving transformations one can associate a three types of unitary representations: Koopman representation, groupoid representation, and uncountable family of quasi-regular representations defined for each orbit of the action. If additionally an element of a group algebra over the field of complex numbers is given then the corresponding operators associated with each of these representations are defined. We show that there is a strong relation between spectra of them (in the form of equality or containment). More information is known in the case when the measure is invariant or the action is Zimmer amenable (hyperfinite). The result has interpretation in the terms of weak containment of unitary representations. We illustrate the use of this result and of the corresponding techniques (based the Schreier graphs approach), and show how to compute the spectrum of the Cayley graph of the first group of intermediate growth constructed by the speaker in 1980. The talk is based on a joint paper with A.Dudko

Supplements
27481?type=thumb Grigorchuck Notes 3.29 MB application/pdf Download
Video/Audio Files

14652

H.264 Video 14652.mp4 318 MB video/mp4 rtsp://videos.msri.org/data/000/027/259/original/14652.mp4 Download
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