224 research outputs found
Relative Cuntz-Krieger algebras of finitely aligned higher-rank graphs
We define the relative Cuntz-Krieger algebras associated to finitely aligned
higher-rank graphs. We prove versions of the gauge-invariant uniqueness theorem
and the Cuntz-Krieger uniqueness theorem for relative Cuntz-Krieger algebras.Comment: 16 page
Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs
We produce a complete descrption of the lattice of gauge-invariant ideals in
for a finitely aligned -graph . We provide a
condition on under which every ideal is gauge-invariant. We give
conditions on under which satisfies the hypotheses of
the Kirchberg-Phillips classification theorem.Comment: 19 pages. Additional references added in Section 8. Title of Section
6 changed from "The lattice structure" to "The lattice order
Preferred traces on C*-algebras of self-similar groupoids arising as fixed points
Recent results of Laca, Raeburn, Ramagge and Whittaker show that any
self-similar action of a groupoid on a graph determines a 1-parameter family of
self-mappings of the trace space of the groupoid C*-algebra. We investigate the
fixed points for these self-mappings, under the same hypotheses that Laca et
al. used to prove that the C*-algebra of the self-similar action admits a
unique KMS state. We prove that for any value of the parameter, the associated
self-mapping admits a unique fixed point, which is in fact a universal
attractor. This fixed point is precisely the trace that extends to a KMS state
on the C*-algebra of the self-similar action.Comment: 12 pages; v2: this version matches the published versio
A dichotomy for groupoid C*-algebras
We study the finite versus infinite nature of C*-algebras arising from etale
groupoids. For an ample groupoid G, we relate infiniteness of the reduced
C*-algebra of G to notions of paradoxicality of a K-theoretic flavor. We
construct a pre-ordered abelian monoid S(G) which generalizes the type
semigroup introduced by R{\o}rdam and Sierakowski for totally disconnected
discrete transformation groups. This monoid reflects the finite/infinite nature
of the reduced groupoid C*-algebra of G. If G is ample, minimal, and
topologically principal, and S(G) is almost unperforated we obtain a dichotomy
between stable finiteness and pure infiniteness for the reduced C*-algebra of
G.Comment: 40 pages. Version 2: Section 9.2 updated to reflect intersection with
earlier results of Suzuki; thanks to Suzuki for alerting us. Proofs of
Proposition 5.2 and Lemma 9.7 simplified using the refinement property
(correcting an oversight in the proof of Proposition 5.2
Non-Commutative Vector Bundles for Non-Unital Algebras
We revisit the characterisation of modules over non-unital -algebras
analogous to modules of sections of vector bundles. A fullness condition on the
associated multiplier module characterises a class of modules which closely
mirror the commutative case. We also investigate the multiplier-module
construction in the context of bi-Hilbertian bimodules, particularly those of
finite numerical index and finite Watatani index
Aperiodicity and cofinality for finitely aligned higher-rank graphs
We introduce new formulations of aperiodicity and cofinality for finitely
aligned higher-rank graphs \Lambda, and prove that C*(\Lambda) is simple if and
only if \Lambda is aperiodic and cofinal. The main advantage of our versions of
aperiodicity and cofinality over existing ones is that ours are stated in terms
of finite paths. To prove our main result, we first characterise each of
aperiodicity and cofinality of \Lambda in terms of the ideal structure of
C*(\Lambda). In an appendix we show how our new cofinality condition simplifies
in a number of special cases which have been treated previously in the
literature; even in these settings, our results are new.Comment: 18 page
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