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

### Group actions on labeled graphs and their C*-algebras

We introduce the notion of the action of a group on a labeled graph and the
quotient object, also a labeled graph. We define a skew product labeled graph
and use it to prove a version of the Gross-Tucker theorem for labeled graphs.
We then apply these results to the $C^*$-algebra associated to a labeled graph
and provide some applications in nonabelian duality.Comment: 18 pages, updated versio

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