Limits of groupoid C*-algebras arising from open covers

TL;DR

This paper investigates the asymptotic behavior of certain continuous-trace C*-algebras constructed from open covers, analyzing their limits and connections to UHF algebras through groupoid and cocycle extensions.

Contribution

It introduces a limit groupoid framework to understand the asymptotics of Raeburn-Taylor algebras and relates these limits to UHF C*-algebras.

Findings

Limit groupoid G and cocycle σ characterize algebra asymptotics.

Raeburn-Taylor algebras form a generalized direct limit.

All UHF C*-algebras can be obtained from this limit construction.

Abstract

I. Raeburn and J. Taylor have constructed continuous-trace C*-algebras with a prescribed Dixmier-Douady class, which also depend on the choice of an open cover of the spectrum. We study the asymptotic behavior of these algebras with respect to certain refinements of the cover and appropriate extension of cocycles. This leads to the analysis of a limit groupoid G and a cocycle \sigma, and the algebra C*(G, \sigma) may be regarded as a generalized direct limit of the Raeburn-Taylor algebras. As a special case, all UHF C*-algebras arise from this limit construction.