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Jonathan Krook: C*-simplicity of discrete groups and étale groupoids

Bachelor thesis presentation

Time: Mon 2022-06-13 10.00 - 11.00

Location: Kräftriket, house 6, room 306

Respondent: Jonathan Krook

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In this thesis, we look at a dynamical characterisation of C*-simplicity of discrete groups as presented in [1], and groupoid C*-algebras defined for etale groupoids as considered in [2]. Background information and a more detailed outline of the thesis is provided in the introduction. The introduction is followed by a chapter providing necessary preliminary information on C*-algebras, where for example the Gelfand representation and the Gelfand-Naimark-Segal construction are covered. Following the article [1], we discuss in the next chapter a dynamic characterisation of C*-simplicity of discrete groups. Here, we also show that a discrete group G is C*-simple if and only if the reduced crossed product C(∂F G) ⋊r G is simple, where ∂F G denotes the Furstenberg boundary associated to G. In the final chapter, we present a definition of groupoid C*-algebras that appears in [2]. We also present a definition of the reduced groupoid C*- algebra, although in a different form from what appears in [2], that applies to ´etale groupoids with totally disconnected locally compact Hausdorff space of units, and show that the reduced groupoid C*-algebra with this definition is a groupoid C*-algebra.