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Aidan Sims

Publications and source records attributed to Aidan Sims.

At least 109 records · Page 6Linked to original sources
arXiv2010

Co-universal C*-algebras associated to generalised graphs

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in N. We focus on semigroups P arising as part of a quasi-lattice ordered group (G,P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*_{min}(Lambda) which is co-universal for partial-isometric representations of Lambda which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent C*_{min}(Lambda) for some (N^2 * N)-graph Lambda.

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arXiv2010

Generalised morphisms of k-graphs: k-morphs

In a number of recent papers, (k+l)-graphs have been constructed from k-graphs by inserting new edges in the last l dimensions. These constructions have been motivated by C*-algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce k-morphs, which provide a systematic unifying framework for these various constructions. We think of k-morphs as the analogue, at the level of k-graphs, of C*-correspondences between C*-algebras. To make this analogy explicit, we introduce a category whose objects are k-graphs and whose morphisms are isomorphism classes of k-morphs. We show how to extend the assignment Λ\mapsto C*(Λ) to a functor from this category to the category whose objects are C*-algebras and whose morphisms are isomorphism classes of C*-correspondences.

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arXiv2010

Simplicity of C*-algebras associated to row-finite locally convex higher-rank graphs

In previous work, the authors showed that the C*-algebra C*(Λ) of a row-finite higher-rank graph Λwith no sources is simple if and only if Λis both cofinal and aperiodic. In this paper, we generalise this result to row-finite higher-rank graphs which are locally convex (but may contain sources). Our main tool is Farthing's "removing sources" construction which embeds a row-finite locally convex higher-rank graph in a row-finite higher-rank graph with no sources in such a way that the associated C*-algebras are Morita equivalent.

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arXiv2010

A direct approach to co-universal algebras associated to directed graphs

We prove directly that if E is a directed graph in which every cycle has an entrance, then there exists a C*-algebra which is co-universal for Toeplitz-Cuntz-Krieger E-families. In particular, our proof does not invoke ideal-structure theory for graph algebras, nor does it involve use of the gauge action or its fixed point algebra.

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arXiv2009

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-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 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.

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arXiv2009

Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C*-algebra, an Exel-Laca algebra, and an ultragraph C*-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C*-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C*-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C*-algebra of a row-finite graph with no sinks.

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arXiv2009

Aperiodicity and cofinality for finitely aligned higher-rank graphs

We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove that C*(Λ) is simple if and only if Λis 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 Λin terms of the ideal structure of C*(Λ). 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.

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arXiv2009

Graphs of $C^*$-correspondences and Fell bundles

We define the notion of a $Λ$-system of $C^*$-correspondences associated to a higher-rank graph $Λ$. Roughly speaking, such a system assigns to each vertex of $Λ$ a $C^*$-algebra, and to each path in $Λ$ a $C^*$-correspondence in a way which carries compositions of paths to balanced tensor products of $C^*$-correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of $C^*$-correspondences to associate a $C^*$-algebra to each $Λ$-system. We then construct a Fell bundle over the path groupoid $\Gg_Λ$ and show that the $C^*$-algebra of the $Λ$-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.

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arXiv2009

C*-algebras associated to product systems of Hilbert bimodules

Let (G,P) be a quasi-lattice ordered group and let X be a compactly aligned product system over P of Hilbert bimodules. Under mild hypotheses we associate to X a C*-algebra which we call the Cuntz-Nica-Pimsner algebra of X. Our construction generalises a number of others: a sub-class of Fowler's Cuntz-Pimsner algebras for product systems of Hilbert bimodules; Katsura's formulation of Cuntz-Pimsner algebras of Hilbert bimodules; the C*-algebras of finitely aligned higher-rank graphs; and Crisp and Laca's boundary quotients of Toeplitz algebras. We show that for a large class of product systems X, the universal representation of X in its Cuntz-Nica-Pimsner algebra is isometric.

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arXiv2008

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph G we construct a directed graph E such that C*(G) is isomorphic to a full corner of C*(E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

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arXiv2008

The co-universal C*-algebra of a row-finite graph

Let E be a row-finite directed graph. We prove that there exists a C*-algebra C*_{min}(E) with the following co-universal property: given any C*-algebra B generated by a Toeplitz-Cuntz-Krieger E-family in which all the vertex projections are nonzero, there is a canonical homomorphism from B onto C*_{min}(E). We also identify when a homomorphism from B to C*_{min}(E) obtained from the co-universal property is injective. When every loop in E has an entrance, C*_{min}(E) coincides with the graph C*-algebra C*(E), but in general, C*_{min}(E) is a quotient of C*(E). We investigate the properties of C*_{min}(E) with emphasis on the utility of co-universality as the defining property of the algebra.

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arXiv2008

C*-algebras associated to coverings of k-graphs

A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C*-algebras. We show how to build a (k+1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k+1)-graph. Our main focus is on computing the K-theory of the (k+1)-graph algebra from that of the component k-graph algebras. Examples of our construction include a realisation of the Kirchberg algebra \mathcal{P}_n whose K-theory is opposite to that of \mathcal{O}_n, and a class of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens algebras.

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arXiv2008

Coverings of skew-products and crossed products by coactions

Consider a projective limit G of finite groups G_n. Fix a compatible family δ^n of coactions of the G_n on a C*-algebra A. From this data we obtain a coaction δof G on A. We show that the coaction crossed product of A by δis isomorphic to a direct limit of the coaction crossed products of A by the δ^n. If A = C*(Λ) for some k-graph Λ, and if the coactions δ^n correspond to skew-products of Λ, then we can say more. We prove that the coaction crossed-product of C*(Λ) by δmay be realised as a full corner of the C*-algebra of a (k+1)-graph. We then explore connections with Yeend's topological higher-rank graphs and their C*-algebras.

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arXiv2008

Ultragraph C*-algebras via topological quivers

Given an ultragraph in the sense of Tomforde, we construct a topological quiver in the sense of Muhly and Tomforde in such a way that the universal C*-algebras associated to the two objects coincide. We apply results of Muhly and Tomforde for topological quiver algebras and of Katsura for topological graph C*-algebras to study the K-theory and gauge-invariant ideal structure of ultragraph C*-algebras.

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arXiv2007

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

We prove that if Λis a row-finite k-graph with no sources, then the associated C^*-algebra is simple if and only if Λis cofinal and satisfies Kumjian and Pask's Condition (A). We prove that Condition (A) is equivalent to a suitably modified version of Robertson and Steger's original nonperiodicity condition (H3) which in particular involves only finite paths. We also characterise both cofinality and aperiodicity of Λin terms of ideals in C^*(Λ).

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arXiv2007

Crossed products of k-graph C*-algebras by Z^l

An action of Z^l by automorphisms of a k-graph induces an action of Z^l by automorphisms of the corresponding k-graph C*-algebra. We show how to construct a (k+l)-graph whose C*-algebra coincides with the crossed product of the original k-graph algebra by Z^l. We then investigate the structure of the crossed-product C*-algebra.

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arXiv2007

Noncommutative manifolds from graph and $k$-graph $C^*$-algebras

In previous papers, we constructed smooth (1,\infty)-summable semfinite spectral triples for graph algebras with a faithful trace, and (k,\infty)-summable semifinite spectral triples for k-graph algebras. In this paper we identify classes of graphs and k-graphs which satisfy a version of Connes' conditions for noncommutative manifolds.

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