234 research outputs found

### 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 $C^*$-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 $K$-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

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