Twisted k-Graph Algebras Associated to Bratteli Diagrams

Abstract

Given a system of coverings of k-graphs, we show that the second cohomology of the resulting (k + 1)-graph is isomorphic to that of any one of the k-graphs in the system, and compute the semifinite traces of the resulting twisted (k + 1)-graph C *-algebras. We then consider Bratteli diagrams of 2-graphs whose twisted C *-algebras are matrix algebras over noncommutative tori. For such systems we calculate the ordered K-theory of the resulting twisted 3-graph C *-algebras. We deduce that every such C *-algebra is Morita equivalent to the C *-algebra of a rank-2 Bratteli diagram in the sense of Pask–Raeburn–Rørdam–Sims.