Simplicity of 2-graph algebras associated to Dynamical Systems

We give a combinatorial description of a family of 2-graphs which subsumes those described by Pask, Raeburn and Weaver. Each 2-graph $\Lambda$ we consider has an associated $C^*$-algebra, denoted $C^*(\Lambda)$, which is simple and purely infinite when $\Lambda$ is aperiodic. We give new, straightforward conditions which ensure that $\Lambda$ is aperiodic. These conditions are highly tractable as we only need to consider the finite set of vertices of $\Lambda$ in order to identify aperiodicity. In addition, the path space of each 2-graph can be realised as a two-dimensional dynamical system which we show must have zero entropy.Comment: 19 page

The Noncommutative Geometry of Graph C∗C^*-Algebras I: The Index Theorem

We investigate conditions on a graph $C^*$-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth $(1,\infty)$-summable semfinite spectral triple. The local index theorem allows us to compute the pairing with $K$-theory. This produces invariants in the $K$-theory of the fixed point algebra, and these are invariants for a finer structure than the isomorphism class of $C^*(E)$.Comment: 33 page

A dual graph construction for higher-rank graphs, and KK-theory for finite 2-graphs

Given a $k$-graph $\Lambda$ and an element $p$ of \NN^k, we define the dual $k$-graph, $p\Lambda$. We show that when $\Lambda$ is row-finite and has no sources, the $C^*$-algebras $C^*(\Lambda)$ and $C^*(p\Lambda)$ coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the $K$-theory of $C^*(\Lambda)$ when $\Lambda$ is finite and strongly connected and satisfies the aperiodicity condition.Comment: 9 page