Aviv Censor

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*( G_U,s), where G_U  is a groupoid corresponding to an open cover U of T, and s is a 2-cocycle of G_U. In this work we explore the asymptotic behavior of the algebras C*( G_U,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 G_n the groupoid corresponding to the cover W⁽ⁿ⁾. The asymptotic behavior of the sequence C*( G_n,s_n), for compatible 2-cocycles s_n, 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*( G_n,s_n) 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*( G_n). In light of this, in a more general setting we regard our construction as a certain generalized direct limit of the sequence {C*( G_n,s_n)}. 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.