Ph.D Thesis
Ph.D Student Aviv Censor Taking Groupoid C*-Algebras to the Limit Department of Mathematics Full Professor Aljadeff Eli Full Professor Solel Baruch

Abstract

Let T be a compact Hausdorff space. In 1985, Raeburn and Taylor introduced a family of continuous trace C*-algebras with the property that any Dixmier-Douady class in H3(T ;Z) can be realized by such an algebra. A Raeburn-Taylor C*-algebra can be presented as a twisted groupoid C*-algebra C*( GU,s), where GU  is a groupoid corresponding to an open cover U of T, and s is a 2-cocycle of GU. In this work we explore the asymptotic behavior of the algebras C*( GU,s), as one refines the covers U.

Let W(0),W(1),W(2),… be a sequence of open covers of T , where W(0) = U and each W(i) is a refinement (of a particular sort) of W(i+1). Denote by Gn the groupoid corresponding to the cover W(n). The asymptotic behavior of the sequence C*( Gn,sn), for compatible 2-cocycles sn, is encoded in the C*-algebra of a groupoid G which we introduce, and a 2-cocycle s of G.

In our main result we construct for every n an isometric *-homomorphism from C*( Gn,sn) to C*( G,s). As a very special case, when T is a point, our construction produces all UHF C*-algebras as algebras of the form C*(G). In this case C*(G) is the direct limit of the algebras C*( Gn). In light of this, in a more general setting we regard our construction as a certain generalized direct limit of the sequence {C*( Gn,sn)}. We also study the properties of the groupoid G and several other groupoids related to covers and to sequences of refinements, which we introduce.

This research was in part joint work with Daniel Markiewicz.