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

C*-algebras associated to coverings of k-graphs

A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C*-algebras. We show how to build a (k+1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k+1)-graph. Our main focus is on computing the K-theory of the (k+1)-graph algebra from that of the component k-graph algebras. Examples of our construction include a realisation of the Kirchberg algebra \mathcal{P}_n whose K-theory is opposite to that of \mathcal{O}_n, and a class of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens algebras.Comment: 44 pages, 2 figures, some diagrams drawn using picTeX. v2. A number of typos corrected, some references updated. The statements of Theorem 6.7(2) and Corollary 6.8 slightly reworded for clarity. v3. Some references updated; in particular, theorem numbering of references to Evans updated to match published versio