48 research outputs found
-algebras and Fell bundles associated to a textile system
The notion of textile system was introduced by M. Nasu in order to analyze
endomorphisms and automorphisms of topological Markov shifts. A textile system
is given by two finite directed graphs and and two morphisms , with some extra properties. It turns out that a textile system determines a
first quadrant two-dimensional shift of finite type, via a collection of Wang
tiles, and conversely, any such shift is conjugate to a textile shift. In the
case the morphisms and have the path lifting property, we prove that
they induce groupoid morphisms between the
corresponding \'etale groupoids of and .
We define two families and of
-algebras associated to a textile shift, and compute them in specific
cases. These are graph algebras, associated to some one-dimensional shifts of
finite type constructed from the textile shift. Under extra hypotheses, we also
define two families of Fell bundles which encode the complexity of these
two-dimensional shifts. We consider several classes of examples of textile
shifts, including the full shift, the Golden Mean shift and shifts associated
to rank two graphs.Comment: 14 pages, 4 figure
Fell bundles associated to groupoid morphisms
Given a continuous open surjective morphism of \'etale
groupoids with amenable kernel, we construct a Fell bundle over and
prove that its C*-algebra is isomorphic to . This is
related to results of Fell concerning C*-algebraic bundles over groups. The
case , a locally compact space, was treated earlier by Ramazan. We
conclude that is strongly Morita equivalent to a crossed product,
the C*-algebra of a Fell bundle arising from an action of the groupoid on a
C*-bundle over . We apply the theory to groupoid morphisms obtained from
extensions of dynamical systems and from morphisms of directed graphs with the
path lifting property. We also prove a structure theorem for abelian Fell
bundles.Comment: 12 pages, revised version, references added; to appear in Mathematica
Scandinavic