We define the categorical cohomology of a k-graph Λ and show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted k-graph C * -algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k-graph C * -algebra is isomorphic to a twisted groupoid C * -algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k-graph C * -algebras are nuclear and belong to the bootstrap class.
We consider the higher-rank graphs introduced by Kumjian and Pask as models for higherrank Cuntz-Krieger algebras. We describe a variant of the Cuntz-Krieger relations which applies to graphs with sources, and describe a local convexity condition which characterizes the higher-rank graphs that admit a non-trivial Cuntz-Krieger family. We then prove versions of the uniqueness theorems and classifications of ideals for the C * -algebras generated by Cuntz-Krieger families.
We generalise the theory of Cuntz-Krieger families and graph algebras to the class of finitely aligned k-graphs. This class contains in particular all row-finite k-graphs. The CuntzKrieger relations for non-row-finite k-graphs look significantly different from the usual ones, and this substantially complicates the analysis of the graph algebra. We prove a gaugeinvariant uniqueness theorem and a Cuntz-Krieger uniqueness theorem for the C Ã -algebras of finitely aligned k-graphs. r 2004 Elsevier Inc. All rights reserved. MSC: primary 46L05 convex or row-finite, and we do allow them to have sources. When k ¼ 1 or the k-graph is row-finite and locally convex, our new Cuntz-Krieger relations are equivalent to the usual ones. We show that for every finitely aligned k-graph L; there is a family of nonzero partial isometries which satisfies the new relations, and we define C Ã ðLÞ to be the universal C Ã -algebra generated by such a family. We then prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for C Ã ðLÞ: Our analysis is elementary in the sense that we do not use groupoids, partial actions or Hilbert bimodules, though we cheerfully acknowledge that we have gained insight from the models these theories provide.The results in this paper extend the existing theory of graph algebras in several directions. Since 1-graphs are always finitely aligned, and our new relations are then equivalent to the usual ones (Proposition B.1), our approach provides the first elementary analysis of the C Ã -algebra of an arbitrary directed graph. Our results are also new for finitely aligned k-graphs without sources; those interested primarily in this situation may mentally replace all the symbols L pn by L n ; and thereby avoid several technical complications. Even for row-finite k-graphs we make significant improvements on the existing theory: for non-locally-convex row-finite k-graphs, our Cuntz-Krieger families may have every vertex projection nonzero, unlike those in [13] (see Example A.1).In Section 2, we describe our new Cuntz-Krieger relations for a finitely aligned kgraph L; define C Ã ðLÞ to be the universal C Ã -algebra generated by a Cuntz-Krieger family, and investigate some of its basic properties. We discuss a notion of boundary paths which we use to construct a Cuntz-Krieger family in which every vertex projection is nonzero.The core in C Ã ðLÞ is the fixed-point algebra C Ã ðLÞ g for the gauge action g of T k : In Section 3, we show that the core is AF, and deduce that a homomorphism p of C Ã ðLÞ which is nonzero at each vertex projection is injective on the core.Our proof that C Ã ðLÞ g is AF is quite different from the argument which we gave for row-finite k-graphs in [13] in that we do not describe C Ã ðLÞ g as a direct limit over N k : Instead, we describe C Ã ðLÞ g as the increasing union of finite-dimensional algebras indexed by finite sets of paths, and produce families of matrix units which span these algebras. In addition to showing that C Ã ðLÞ is AF, this formu...
We prove that the full C * -algebra of a second-countable, Hausdorff,étale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G has totally disconnected unit space, then the complex * -algebra of its inverse semigroup of compact open bisections, as introduced by Steinberg, is simple if and only if G is both effective and minimal.
Let (G, P) be a quasi‐lattice ordered group, and let X be a product system over P of Hilbert bimodules. Under mild hypotheses, we associate to X a C*‐algebra which is co‐universal for injective Nica covariant Toeplitz representations of X which preserve the gauge coaction. Under appropriate amenability criteria, this co‐universal C*‐algebra coincides with the Cuntz‐Nica‐Pimsner algebra introduced by Sims and Yeend. We prove two key uniqueness theorems, and indicate how to use our theorems to realize a number of reduced crossed products as instances of our co‐universal algebras. In each case, it is an easy corollary that the Cuntz‐Nica‐Pimsner algebra is isomorphic to the corresponding full crossed product.
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