February 1, 2000
Graphs as categories, Graph algebra, Path groupoid, C*-algebra
Primary 46L05; Secondary 46L55.
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
Department of Mathematics (084), University of Nevada, Reno NV 89557-0045, USA.
Department of Mathematics, University of Newcastle, NSW 2308, Australia