ORCID Profile
0000-0002-1965-6451
Current Organisation
University of Wollongong
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In Research Link Australia (RLA), "Research Topics" refer to ANZSRC FOR and SEO codes. These topics are either sourced from ANZSRC FOR and SEO codes listed in researchers' related grants or generated by a large language model (LLM) based on their publications.
Pure Mathematics | Operator Algebras and Functional Analysis | Functional Analysis | Category Theory, K Theory, Homological Algebra | Category Theory, K Theory, Homological Algebra | Dynamical Systems | Group Theory and Generalisations | Topology | Algebra and Number Theory
Expanding Knowledge in the Mathematical Sciences | Mathematical sciences |
Publisher: Springer Netherlands
Date: 2001
Publisher: Institute of Mathematics, Polish Academy of Sciences
Date: 2008
DOI: 10.4064/SM187-2-3
Publisher: Elsevier BV
Date: 09-2012
Publisher: American Mathematical Society (AMS)
Date: 29-06-2006
DOI: 10.1090/S0002-9939-05-07994-3
Abstract: Given a k k -graph Λ \\Lambda and an element p p of N k \\mathbb {N}^k , we define the dual k k -graph, p Λ p\\Lambda . We show that when Λ \\Lambda is row-finite and has no sources, the C ∗ C^* -algebras C ∗ ( Λ ) C^*(\\Lambda ) and C ∗ ( p Λ ) C^*(p\\Lambda ) coincide. We use this isomorphism to apply Robertson and Steger’s results to calculate the K K -theory of C ∗ ( Λ ) C^*(\\Lambda ) when Λ \\Lambda is finite and strongly connected and satisfies the aperiodicity condition.
Publisher: Springer Science and Business Media LLC
Date: 03-2015
Publisher: Elsevier BV
Date: 09-2009
Publisher: Cambridge University Press (CUP)
Date: 04-10-2021
DOI: 10.1017/S0013091521000626
Abstract: We study the structure and compute the stable rank of $C^{*}$ -algebras of finite higher-rank graphs. We completely determine the stable rank of the $C^{*}$ -algebra when the $k$ -graph either contains no cycle with an entrance or is cofinal. We also determine exactly which finite, locally convex $k$ -graphs yield unital stably finite $C^{*}$ -algebras. We give several ex les to illustrate our results.
Publisher: Springer Science and Business Media LLC
Date: 26-09-2009
Publisher: Elsevier BV
Date: 08-2016
Publisher: Indiana University Mathematics Journal
Date: 2006
Publisher: Elsevier BV
Date: 03-2011
Publisher: Springer Science and Business Media LLC
Date: 20-09-2013
Publisher: Springer Science and Business Media LLC
Date: 29-01-2015
Publisher: Wiley
Date: 04-2007
DOI: 10.1112/BLMS/BDM006
Publisher: Elsevier BV
Date: 07-2015
Publisher: Elsevier BV
Date: 10-2018
Publisher: Springer Science and Business Media LLC
Date: 07-2009
Publisher: Elsevier BV
Date: 06-2015
Publisher: Cambridge University Press (CUP)
Date: 13-08-2013
DOI: 10.1017/S001708951300044X
Abstract: We construct a representation of each finitely aligned aperiodic k -graph Λ on the Hilbert space $\\mathcal{H}^{\\rm ap}$ with basis indexed by aperiodic boundary paths in Λ. We show that the canonical expectation on $\\mathcal{B}(\\mathcal{H}^{\\rm ap})$ restricts to an expectation of the image of this representation onto the subalgebra spanned by the final projections of the generating partial isometries. We then show that every quotient of the Toeplitz algebra of the k -graph admits an expectation compatible with this one. Using this, we prove that the image of our representation, which is canonically isomorphic to the Cuntz–Krieger algebra, is co-universal for Toeplitz–Cuntz–Krieger families consisting of non-zero partial isometries.
Publisher: Elsevier BV
Date: 09-2013
Publisher: Elsevier BV
Date: 02-2015
Publisher: Springer International Publishing
Date: 25-10-2016
Publisher: Wiley
Date: 16-03-2011
DOI: 10.1112/PLMS/PDQ028
Publisher: Cambridge University Press (CUP)
Date: 21-12-2008
DOI: 10.1017/IS007011015JKT013
Abstract: This paper is comprised of two related parts. First we discuss which k -graph algebras have faithful traces. We characterise the existence of a faithful semifinite lower-semicontinuous gauge-invariant trace on C * (Λ) in terms of the existence of a faithful graph trace on Λ. Second, for k -graphs with faithful gauge invariant trace, we construct a smooth ( k ,∞)-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K -theory. This numerical pairing can be obtained by applying the trace to a KK -pairing with values in the K -theory of the fixed point algebra of the T k action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.
Publisher: Springer Science and Business Media LLC
Date: 21-03-2016
Publisher: Cambridge University Press (CUP)
Date: 30-04-2013
DOI: 10.1017/S0013091512000338
Abstract: Results of Fowler and Sims show that every k -graph is completely determined by its k -coloured skeleton and collection of commuting squares. Here we give an explicit description of the k -graph associated with a given skeleton and collection of squares and show that two k -graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k -graph Λ is 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 characterization, originally due to Robertson and Sims, of simplicity of the C *-algebra of a row-finite k -graph with no sources.
Publisher: American Mathematical Society (AMS)
Date: 03-03-2015
DOI: 10.1090/S0002-9947-2015-06209-6
Abstract: We define the categorical cohomology of a k k -graph Λ \\Lambda 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 k -graph C ∗ C^* -algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k k -graph C ∗ C^* -algebra is isomorphic to a twisted groupoid C ∗ C^* -algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k k -graph C ∗ C^* -algebras are nuclear and belong to the bootstrap class.
Publisher: New Zealand Journal of Mathematics Committee
Date: 19-09-2021
DOI: 10.53733/136
Abstract: We analyse extensions $\\Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $\\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of ex les and applications.
Publisher: Springer Science and Business Media LLC
Date: 03-04-2014
Publisher: Cambridge University Press (CUP)
Date: 27-01-2003
DOI: 10.1017/S0013091501000645
Abstract: We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank 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. AMS 2000 Mathematics subject classification: Primary 46L05
Publisher: Elsevier BV
Date: 10-2016
Publisher: Elsevier BV
Date: 02-2014
Publisher: Elsevier BV
Date: 07-2012
Publisher: Indiana University Mathematics Journal
Date: 2010
Publisher: Cambridge University Press (CUP)
Date: 17-04-2019
DOI: 10.1017/ETDS.2019.20
Abstract: We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.
Publisher: Cambridge University Press (CUP)
Date: 11-08-2014
DOI: 10.1017/ETDS.2014.52
Abstract: We consider the dynamics on the $C^{\\ast }$ -algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani [KMS states for gauge action on ${\\mathcal{O}}_{A}$ . Math. Japon. 29 (1984), 607–619] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo–Martin–Schwinger (KMS) state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of ex les. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph.
Publisher: Canadian Mathematical Society
Date: 12-2006
Abstract: We produce a complete description of the lattice of gauge-invariant ideals in C *(Λ) for a finitely aligned k -graph Λ. We provide a condition on Λ under which every ideal is gauge-invariant. We give conditions on Λ under which C *(Λ) satisfies the hypotheses of the Kirchberg–Phillips classification theorem.
Publisher: American Mathematical Society (AMS)
Date: 30-11-2001
DOI: 10.1090/S0002-9947-01-02911-7
Abstract: We build upon Mac Lane’s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid G \\mathcal G . We show that the existing notions of product systems fit into our categorical framework, as do the k 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 \\mathcal G . In particular, we obtain necessary and sufficient conditions under which a collection of k k 1 1 -graphs form the coordinate graphs of a k k -graph.
Publisher: Walter de Gruyter GmbH
Date: 2010
Publisher: Cambridge University Press (CUP)
Date: 10-05-2010
DOI: 10.1017/S0305004110000034
Abstract: 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 Λ 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 Λ in 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.
Publisher: Cambridge University Press (CUP)
Date: 17-12-2019
Publisher: Elsevier BV
Date: 2014
Publisher: Springer Science and Business Media LLC
Date: 26-09-2015
Publisher: World Scientific Pub Co Pte Lt
Date: 13-03-2017
DOI: 10.1142/S1793525317500108
Abstract: For bi-Hilbertian [Formula: see text]-bimodules, in the sense of Kajiwara–Pinzari–Watatani, we construct a Kasparov module representing the extension class defining the Cuntz–Pimsner algebra. The construction utilises a singular expectation which is defined using the [Formula: see text]-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz–Pimsner algebras.
Publisher: Cambridge University Press (CUP)
Date: 02-10-2023
DOI: 10.1017/PRM.2023.99
Publisher: Elsevier BV
Date: 04-0003
Publisher: Cambridge University Press (CUP)
Date: 06-2009
DOI: 10.1017/S144678870800030X
Abstract: 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 δ of G on A . We show that the coaction crossed product of A by δ is 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 δ may be realized 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.
Publisher: Elsevier BV
Date: 05-2013
Publisher: Cambridge University Press (CUP)
Date: 10-06-2015
DOI: 10.1017/S0013091515000061
Abstract: We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ , this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k -graphs: projective limits and crossed products by finitely generated free abelian groups.
Publisher: Elsevier BV
Date: 10-2006
Publisher: Elsevier BV
Date: 07-2016
Publisher: Springer Science and Business Media LLC
Date: 14-11-2013
Publisher: American Mathematical Society (AMS)
Date: 20-12-2011
DOI: 10.1090/S0002-9947-2010-05152-9
Abstract: In a number of recent papers, ( k + l ) (k+l) -graphs have been constructed from k k -graphs by inserting new edges in the last l l dimensions. These constructions have been motivated by C ∗ C^* -algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce k k -morphs, which provide a systematic unifying framework for these various constructions. We think of k k -morphs as the analogue, at the level of k k -graphs, of C ∗ C^* -correspondences between C ∗ C^* -algebras. To make this analogy explicit, we introduce a category whose objects are k k -graphs and whose morphisms are isomorphism classes of k k -morphs. We show how to extend the assignment Λ ↦ C ∗ ( Λ ) \\Lambda \\mapsto C^*(\\Lambda ) to a functor from this category to the category whose objects are C ∗ C^* -algebras and whose morphisms are isomorphism classes of C ∗ C^* -correspondences.
Publisher: Elsevier BV
Date: 08-2004
Publisher: Elsevier BV
Date: 08-2019
Start Date: 04-2005
End Date: 12-2008
Amount: $220,851.00
Funder: Australian Research Council
View Funded ActivityStart Date: 07-2022
End Date: 06-2025
Amount: $384,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2012
End Date: 12-2015
Amount: $330,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2012
End Date: 12-2015
Amount: $435,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 09-2018
End Date: 03-2022
Amount: $347,167.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2011
End Date: 12-2014
Amount: $562,027.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2009
End Date: 12-2012
Amount: $255,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2009
End Date: 12-2012
Amount: $186,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 05-2007
End Date: 09-2010
Amount: $279,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2015
End Date: 12-2017
Amount: $345,300.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2015
End Date: 12-2018
Amount: $310,700.00
Funder: Australian Research Council
View Funded ActivityStart Date: 06-2020
End Date: 09-2023
Amount: $461,000.00
Funder: Australian Research Council
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