Higher rank graph C*-algebras

TL;DR

This paper extends the theory of C*-algebras by associating them with higher rank graphs, establishing their properties, and exploring their structural and dynamical aspects.

Contribution

It introduces a new framework linking higher rank graphs to C*-algebras, including conditions for simplicity, pure infiniteness, and AF properties, and develops methods for constructing such graphs.

Findings

C*-algebras associated with higher rank graphs are isomorphic to those of their path groupoids.

Conditions for the C*-algebra to be simple, purely infinite, and AF are identified.

A technique for constructing rank 2 graphs from commuting rank 1 graphs is presented.

Abstract

Building on recent work of Robertson and Steger, we associate a C*-algebra to a combinatorial object which may be thought of as a higher rank graph. This C*-algebra is shown to be isomorphic to that of the associated path groupoid. Sufficient conditions on the higher rank graph are found for the associated C*-algebra to be simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from ``commuting'' rank 1 graphs is given.