440 research outputs found
Spectral triples for hyperbolic dynamical systems
Spectral triples are defined for C*-algebras associated with hyperbolic
dynamical systems known as Smale spaces. The spectral dimension of one of these
spectral triples is shown to recover the topological entropy of the Smale
space
C*-algebras of tilings with infinite rotational symmetry
A tiling with infinite rotational symmetry, such as the Conway-Radin Pinwheel
Tiling, gives rise to a topological dynamical system to which an \'etale
equivalence relation is associated. A groupoid C*-algebra for a tiling is
produced and a separating dense set is exhibited in the C*-algebra which
encodes the structure of the topological dynamical system. In the case of a
substitution tiling, natural subsets of this separating dense set are used to
define an AT-subalgebra of the C*-algebra. Finally our results are applied to
the Pinwheel Tiling
K-Theoretic Duality for Hyperbolic Dynamical Systems
The K-theoretic analog of Spanier-Whitehead duality for noncommutative
C*-algebras is shown to hold for the Ruelle algebras associated to irreducible
Smale spaces. This had previously been proved only for shifts of finite type.
Implications of this result as well as relations to the Baum-Connes conjecture
and other topics are also considered.Comment: 36 page
Twisted C*-algebras associated to finitely aligned higher-rank graphs
We introduce twisted relative Cuntz-Krieger algebras associated to finitely
aligned higher-rank graphs and give a comprehensive treatment of their
fundamental structural properties. We establish versions of the usual
uniqueness theorems and the classification of gauge-invariant ideals. We show
that all twisted relative Cuntz-Krieger algebras associated to finitely aligned
higher-rank graphs are nuclear and satisfy the UCT, and that for twists that
lift to real-valued cocycles, the K-theory of a twisted relative Cuntz-Krieger
algebra is independent of the twist. In the final section, we identify a
sufficient condition for simplicity of twisted Cuntz-Krieger algebras
associated to higher-rank graphs which are not aperiodic. Our results indicate
that this question is significantly more complicated than in the untwisted
setting.Comment: Version 2: This paper has now appeared in Documenta Mathematica. This
version on arXiv exactly matches the pagination and format of the published
version. Original published version available from
http://www.math.uni-bielefeld.de/documenta/vol-19/28.htm
Graph algebras and orbit equivalence
We introduce the notion of orbit equivalence of directed graphs, following
Matsumoto's notion of continuous orbit equivalence for topological Markov
shifts. We show that two graphs in which every cycle has an exit are orbit
equivalent if and only if there is a diagonal-preserving isomorphism between
their -algebras. We show that it is necessary to assume that every cycle
has an exit for the forward implication, but that the reverse implication holds
for arbitrary graphs. As part of our analysis of arbitrary graphs we
construct a groupoid from the graph
algebra and its diagonal subalgebra which generalises
Renault's Weyl groupoid construction applied to . We
show that recovers the graph groupoid
without the assumption that every cycle in has an exit,
which is required to apply Renault's results to . We
finish with applications of our results to out-splittings of graphs and to
amplified graphs.Comment: 27 page
Zappa-Sz\'ep products of semigroups and their C*-algebras
Zappa-Sz\'ep products of semigroups encompass both the self-similar group
actions of Nekrashevych and the quasi-lattice-ordered groups of Nica. We use
Li's construction of semigroup -algebras to associate a -algebra to
Zappa-Sz\'ep products and give an explicit presentation of the algebra. We then
define a quotient -algebra that generalises the Cuntz-Pimsner algebras for
self-similar actions. We indicate how known examples, previously viewed as
distinct classes, fit into our unifying framework. We specifically discuss the
Baumslag-Solitar groups, the binary adding machine, the semigroup
, and the -semigroup
Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions
We consider a family of Cuntz-Pimsner algebras associated to self-similar
group actions, and their Toeplitz analogues. Both families carry natural
dynamics implemented by automorphic actions of the real line, and we
investigate the equilibrium states (the KMS states) for these dynamical
systems.
We find that for all inverse temperatures above a critical value, the KMS
states on the Toeplitz algebra are given, in a very concrete way, by traces on
the full group algebra of the group. At the critical inverse temperature, the
KMS states factor through states of the Cuntz-Pimsner algebra; if the
self-similar group is contracting, then the Cuntz-Pimsner algebra has only one
KMS state. We apply these results to a number of examples, including the
self-similar group actions associated to integer dilation matrices, and the
canonical self-similar actions of the basilica group and the Grigorchuk group.Comment: The paper has been updated to agree with the published versio
Functorial properties of Putnam's homology theory for Smale spaces
We investigate functorial properties of Putnam's homology theory for Smale
spaces. Our analysis shows that the addition of a conjugacy condition is
necessary to ensure functoriality. Several examples are discussed that
elucidate the need for our additional hypotheses. Our second main result is a
natural generalization of Putnam's Pullback Lemma from shifts of finite type to
non-wandering Smale spaces.Comment: Updated to agree with published versio
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