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 we consider
has an associated -algebra, denoted , which is simple and
purely infinite when is aperiodic. We give new, straightforward
conditions which ensure that is aperiodic. These conditions are
highly tractable as we only need to consider the finite set of vertices of
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 -Algebras I: The Index Theorem
We investigate conditions on a graph -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 -summable
semfinite spectral triple. The local index theorem allows us to compute the
pairing with -theory. This produces invariants in the -theory of the
fixed point algebra, and these are invariants for a finer structure than the
isomorphism class of .Comment: 33 page
A dual graph construction for higher-rank graphs, and -theory for finite 2-graphs
Given a -graph and an element of \NN^k, we define the dual
-graph, . We show that when is row-finite and has no
sources, the -algebras and coincide. We use
this isomorphism to apply Robertson and Steger's results to calculate the
-theory of when 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 -algebra associated to a labeled graph
and provide some applications in nonabelian duality.Comment: 18 pages, updated versio
- β¦