arXiv subjects

David Pask

Publications and source records attributed to David Pask.

At least 37 records · Page 2Linked to original sources

arXiv2016

Topological spaces associated to higher-rank graphs

We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that identifying isomorphic co-hereditary subgraphs in a disjoint union of two rank-$k$ graphs gives rise to pullbacks of the associated $C^*$-algebras. We describe a combinatorial version of the connected-sum operation and apply it to the rank-2-graph realisations of the four basic surfaces to deduce that every compact 2-manifold is the topological realisation of a rank-2 graph. We also show how to construct $k$-spheres and wedges of $k$-spheres as topological realisations of rank-$k$ graphs.

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arXiv2016

Real rank and topological dimension of higher rank graph algebras

We study dimension theory for the $C^*$-algebras of row-finite $k$-graphs with no sources. We establish that strong aperiodicity - the higher-rank analogue of condition (K) - for a $k$-graph is necessary and sufficient for the associated $C^*$-algebra to have topological dimension zero. We prove that a purely infinite $2$-graph algebra has real-rank zero if and only if it has topological dimension zero and satisfies a homological condition that can be characterised in terms of the adjacency matrices of the $2$-graph. We also show that a $k$-graph $C^*$-algebra with topological dimension zero is purely infinite if and only if all the vertex projections are properly infinite. We show by example that there are strongly purely infinite $2$-graphs algebras, both with and without topological dimension zero, that fail to have real-rank zero.

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arXiv2016

C*-algebras associated to graphs of groups

To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C*-algebras associated to generalised Baumslag-Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.

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arXiv2015

$C^*$-algebras of labelled graphs III - $K$-theory computations

In this paper we give a formula for the $K$-theory of the $C^*$-algebra of a weakly left-resolving labelled space. This is done by realising the $C^*$-algebra of a weakly left-resolving labelled space as the Cuntz-Pimsner algebra of a $C^*$-correspondence. As a corollary we get a gauge invariant uniqueness theorem for the $C^*$-algebra of any weakly left-resolving labelled space. In order to achieve this we must modify the definition of the $C^*$-algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of $C^*$-algebras which are associated with shift spaces and labelled graph algebras. Hence, by computing the $K$-theory of a labelled graph algebra we are providing a common framework for computing the $K$-theory of graph algebras, ultragraph algebras, Exel-Laca algebras, Matsumoto algebras and the $C^*$-algebras of Carlsen. We provide an inductive limit approach for computing the $K$-groups of an important class of labelled graph algebras, and give examples.

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arXiv2014

Crossed products and twisted $k$-graph algebras

An automorphism $β$ of a $k$-graph $Λ$ induces a crossed product $C^* ( Λ) \rtimes_β\mathbb{Z}$ which is isomorphic to a $(k+1)$-graph algebra $C^* ( Λ\times_β\mathbb{Z})$. In this paper we show how this process interacts with $k$-graph $C^*$-algebras which have been twisted by an element of their second cohomology group. This analysis is done using a long exact sequence in cohomology associated to this data. We conclude with some examples

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arXiv2014

Twisted k-graph algebras associated to Bratteli diagrams

Given a system of coverings of k-graphs, we show that the cohomology of the resulting (k+1)-graph is isomorphic to that of any one of the k-graphs in the system. We then consider Bratteli diagrams of 2-graphs whose twisted C*-algebras are matrix algebras over noncommutative tori. For such systems we calculate the ordered K-theory and the gauge-invariant semifinite traces of the resulting 3-graph C*-algebras. We deduce that every simple C*-algebra of this form is Morita equivalent to the C*-algebra of a rank-2 Bratteli diagram in the sense of Pask-Raeburn-Rørdam-Sims.

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arXiv2013

On twisted higher-rank graph C*-algebras

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.

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arXiv2013

Simplicity of the C*-algebras of skew product k-graphs

We consider conditions on a $k$-graph $Λ$, a semigroup $S$ and a functor $η: Λ\to S$ which ensure that the $C^*$-algebra of the skew-product graph $Λ\times_ηS$ is simple. Our results allow give some necessary and sufficient conditions for the AF-core of a $k$-graph $C^{*}$-algebra to be simple.

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arXiv2013

Group actions on labeled graphs and their C*-algebras

We introduce the notion of the action of a group on a labeled graph and the quotient object, also a labeled graph. We define a skew product labeled graph and use it to prove a version of the Gross-Tucker theorem for labeled graphs. We then apply these results to the $C^*$-algebra associated to a labeled graph and provide some applications in nonabelian duality.

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arXiv2013

Skew-products of higher-rank graphs and crossed products by semigroups

We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the $C^*$-algebra of the graph. We show that the crossed product by this action is stably isomorphic to the $C^*$-algebra of a quotient graph. Our main tool is Laca's dilation theory for endomorphic actions of Ore semigroups on $C^*$-algebras, which embeds such an action in an automorphic action of the enveloping group on a larger $C^*$-algebra.

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arXiv2012

On the K-theory of twisted higher-rank-graph C*-algebras

We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras. We show that each 2-cocycle on a higher-rank graph taking values in an abelian group determines a continuous bundle of twisted higher-rank graph algebras over the dual group. We use this to show that for a circle-valued 2-cocycle on a higher-rank graph obtained by exponentiating a real-valued cocycle, the K-theory of the twisted higher-rank graph algebra coincides with that of the untwisted one.

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arXiv2012

Aperiodicity and the primitive ideal space of a row-finite $k$-graph $C^*$-algebra

We describe the primitive ideal space of the $C^{\ast}$-algebra of a row-finite $k$-graph with no sources when every ideal is gauge invariant. We characterize which spectral spaces can occur, and compute the primitive ideal space of two examples. In order to do this we prove some new results on aperiodicity. Our computations indicate that when every ideal is gauge invariant, the primitive ideal space only depends on the 1-skeleton of the $k$-graph in question.

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arXiv2012

An elementary approach to C*-algebras associated to topological graphs

We develop notions of a representation of a topological graph E and of a covariant representation of a topological graph E which do not require the machinery of C*-correspondences and Cuntz-Pimsner algebras. We show that the C*-algebra generated by a universal representation of E coincides with the Toeplitz algebra of Katsura's topological-graph bimodule, and that the C*-algebra generated by a universal covariant representation of E coincides with Katsura's topological graph C*-algebra. We exhibit our results by constructing the isomorphism between the C*-algebra of a row-finite directed graph E with no sources and the C*-algebra of the topological graph arising from the shift map acting on infinite path space E^\infty.

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arXiv2011

Homology for higher-rank graphs and twisted C*-algebras

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set \tilde{Q}(Λ) from a k-graph Λ and demonstrate that the homology and topological realisation of Λ coincide with those of \tilde{Q}(Λ) as defined by Grandis.

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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 2-graph algebras associated to Dynamical Systems

We give a combinatorial description of a family of 2-graphs which subsumes those described by Pask, Raeburn and Weaver. Each 2-graph $Λ$ we consider has an associated $C^*$-algebra, denoted $C^*(Λ)$, which is simple and purely infinite when $Λ$ is aperiodic. We give new, straightforward conditions which ensure that $Λ$ is aperiodic. These conditions are highly tractable as we only need to consider the finite set of vertices of $Λ$ in order to identify aperiodicity. In addition, the path space of each 2-graph can be realised as a two-dimensional dynamical system which we show must have zero entropy.

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arXiv2009

Periodic 2-graphs arising from subshifts

Higher-rank graphs were introduced by Kumjian and Pask to provide models for higher-rank Cuntz-Krieger algebras. In a previous paper, we constructed 2-graphs whose path spaces are rank-two subshifts of finite type, and showed that this construction yields aperiodic 2-graphs whose $C^*$-algebras are simple and are not ordinary graph algebras. Here we show that the construction also gives a family of periodic 2-graphs which we call \emph{domino graphs}. We investigate the combinatorial structure of domino graphs, finding interesting points of contact with the existing combinatorial literature, and prove a structure theorem for the $C^*$-algebras of domino graphs.

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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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