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Groupoids: C*-algebras, Rapid Decay and Amenability.

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Title:Groupoids: C*-algebras, Rapid Decay and Amenability.
Authors:Robertson, John C.
Contributors:Mathematics (department)
Date Issued:Aug 2018
Publisher:University of Hawaiʻi at Mānoa
Abstract:In this paper we study properties of groupoids by looking at their C-algebras. We
introduce a notion of rapid decay for transformation groupoids and we show that this is
equivalent to the underlying group having the property of rapid decay. We show that our
definition is equivalent to a number of other properties which are in direct correspondence to
the group case. Additionally, given two bilipschitz equivalent discrete groups we construct
an isomorphism of the corresponding transformation groupoids and are able to reformulate
the open problem of showing invariance of rapid decay under quasi-isometry.
We then begin to examine various notions of amenability when abstracted to measured
etale groupoids. In the group case, the following properties are equivalent:
1) G is amenable
2) Cr (G) = C(G)
3) The trivial representation decends from C(G) to Cr (G).
In the groupoid case we have 1) 2) 3), but it is shown in [19] that Cr (G) = C(G) is not
enough in general to give amenability of G. In this paper we study property 3) for groupoids,
formulate some equivalent statements and show that 3) 2) is also false in general.
Description:Ph.D. Thesis. University of Hawaiʻi at Mānoa 2018.
URI:http://hdl.handle.net/10125/62526
Rights:All UHM dissertations and theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission from the copyright owner.
Appears in Collections: Ph.D. - Mathematics


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