Twisted k-graph algebras associated to Bratteli diagrams

TL;DR

This paper explores the cohomology of k-graphs derived from coverings, analyzes their associated twisted C*-algebras, and computes K-theory and traces, revealing Morita equivalences with certain rank-2 Bratteli diagram algebras.

Contribution

It establishes cohomology isomorphisms for (k+1)-graphs from k-graph systems and computes invariants for their twisted C*-algebras, linking them to known algebra classes.

Findings

Cohomology of (k+1)-graphs matches that of k-graphs in the system.

Twisted C*-algebras are matrix algebras over noncommutative tori.

Simple C*-algebras are Morita equivalent to rank-2 Bratteli diagram algebras.

Abstract

Given a system of coverings of k-graphs, we show that the cohomology of the resulting (k+1)-graph is isomorphic to that of any one of the k-graphs in the system. 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 and the gauge-invariant semifinite traces of the resulting 3-graph C*-algebras. We deduce that every simple C*-algebra of this form is Morita equivalent to the C*-algebra of a rank-2 Bratteli diagram in the sense of Pask-Raeburn-R{\o}rdam-Sims.