Higher Rank Graph C*-Algebras

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. Various results in this paper give sufficient conditions on the higher rank graph 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.

Research of the first author partially supported by NSF grant DMS-9706982
Research of the second author supported by University of Newcastle RMC project grant

Alex Kumjian:
Department of Mathematics (084), University of Nevada, Reno NV 89557-0045, USA.
alex@unr.edu
http://equinox.comnet.unr.edu/homepage/alex/

David Pask:
Department of Mathematics, University of Newcastle, NSW 2308, Australia
davidp@maths.newcastle.edu.au
http://maths.newcastle.edu.au/~davidp/