Aidan Sims

Higher-rank graphs and their C*-algebras

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.

Simplicity of C*-algebras associated to higher-rank graphs

We prove that ifis a row-finite k-graph with no sources, then the associated C � -algebra is simple if and only ifis cofinal and satisfies Kumjian and Pasks aperiodicity condition, known as Condition (A). We prove that the aperiodicity condition is equivalent to a suitably modified version of Robertson and Stegers original nonperiodicity condition (H3) which in particular involves only finite paths. We also characterise both cofinality and aperiodicity ofin terms of ideals in C � (�).

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 C � (�) for a finitely aligned k-graph �. We provide a condition on � under which every ideal is gauge-invariant. We give conditions onunder which C � (�) satisfies the hypotheses of the Kirchberg-Phillips classification theorem.

Product systems over right-angled Artin semigroups

We build upon Mac Lanes definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid G. We show that the existing notions of product systems fit into our categorical framework, as do the k-graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid G. In particular, we obtain necessary and sufficient conditions under which a collection of k 1-graphs form the coordinate graphs of a k-graph.

Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Results of Fowler and Sims show that every k-graph is completely deter- mined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated to a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomor- phism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graphis isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterisation, originally due to Robertson and Sims, of simplicity of the C � -algebra of a row-finite k-graph with no sources.

The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources ☆

Abstract We catalogue the primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources. Each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph; the primitive ideals in the Cuntz–Krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group. The Cuntz–Krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic.

CO-UNIVERSAL ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS, AND GAUGE-INVARIANT UNIQUENESS THEOREMS

Let X be a product system over a quasi-lattice ordered group. 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-universalC � -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 realise 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.

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.

On twisted higher-rank graph C*-algebras

We define the categorical cohomology of a k-graphand 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.

Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras

We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matuis notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras