ORCID Profile
0000-0002-5850-061X
Current Organisation
University of Wollongong
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Pure Mathematics | Operator Algebras and Functional Analysis | Functional Analysis | Category Theory, K Theory, Homological Algebra | Topology And Manifolds | Group Theory and Generalisations | Geometry | Category Theory, K Theory, Homological Algebra | Algebra and Number Theory
Expanding Knowledge in the Mathematical Sciences | Mathematical sciences |
Publisher: Indiana University Mathematics Journal
Date: 2017
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: 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: Elsevier BV
Date: 07-2005
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: Springer Science and Business Media LLC
Date: 10-1991
DOI: 10.1007/BF02782848
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: Springer Science and Business Media LLC
Date: 29-01-2015
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 1996
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 27-07-2018
DOI: 10.4171/JNCG/292
Publisher: Elsevier BV
Date: 03-1997
Publisher: Springer Science and Business Media LLC
Date: 27-08-2013
Publisher: Elsevier BV
Date: 05-2001
Publisher: Royal Society of Chemistry (RSC)
Date: 2021
DOI: 10.1039/D1OB01545A
Abstract: The acridinone 1,9-bis(thio)urea scaffold was repurposed for application in anion transport by appending a variety of electron-withdrawing groups to the peripheral phenyl moieties. High levels of activity were achieved which facilitated strictly electroneutral transport.
Publisher: Cambridge University Press (CUP)
Date: 12-03-2009
DOI: 10.1017/S0143385708000795
Abstract: Higher-rank graphs (or k -graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz–Krieger C * -algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the C * -algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their K -theory. We find ex les whose C * -algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic to the C * -algebras of ordinary directed graphs.
Publisher: Elsevier BV
Date: 10-2016
Publisher: Springer Science and Business Media LLC
Date: 30-09-2011
Publisher: Cambridge University Press (CUP)
Date: 06-10-2015
DOI: 10.1017/ETDS.2015.62
Abstract: In this paper we give a formula for the $K$ -theory of the $C^{\\ast }$ -algebra of a weakly left-resolving labelled space. This is done by realizing the $C^{\\ast }$ -algebra of a weakly left-resolving labelled space as the Cuntz–Pimsner algebra of a $C^{\\ast }$ -correspondence. As a corollary, we obtain a gauge-invariant uniqueness theorem for the $C^{\\ast }$ -algebra of any weakly left-resolving labelled space. In order to achieve this, we must modify the definition of the $C^{\\ast }$ -algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of $C^{\\ast }$ -algebras that 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 $C^{\\ast }$ -algebras of Carlsen. We provide an inductive limit approach for computing the $K$ -groups of an important class of labelled graph algebras, and give ex les.
Publisher: World Scientific Pub Co Pte Lt
Date: 07-2001
Publisher: Indiana University Mathematics Journal
Date: 2010
Publisher: Cambridge University Press (CUP)
Date: 12-1999
DOI: 10.1017/S0143385799151940
Abstract: Given a free action of a group $G$ on a directed graph $E$ we show that the crossed product of $C^* (E)$, the universal $C^*$-algebra of $E$, by the induced action is strongly Morita equivalent to $C^* (E/G)$. Since every connected graph $E$ may be expressed as the quotient of a tree $T$ by an action of a free group $G$ we may use our results to show that $C^* (E)$ is strongly Morita equivalent to the crossed product $C_0 ( \\partial T ) \\times G$, where $\\partial T$ is a certain zero-dimensional space canonically associated to the tree.
Publisher: European Mathematical Society - EMS - Publishing House GmbH
Date: 2016
DOI: 10.4171/JNCG/241
Publisher: Elsevier BV
Date: 2021
Publisher: Informa UK Limited
Date: 12-06-2019
Publisher: Cambridge University Press (CUP)
Date: 08-2003
Publisher: Springer Science and Business Media LLC
Date: 05-02-2017
Publisher: Elsevier BV
Date: 04-2006
Publisher: Springer Science and Business Media LLC
Date: 02-1990
DOI: 10.1007/BF02764730
Publisher: Elsevier BV
Date: 05-2013
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: 10-2006
Publisher: World Scientific Pub Co Pte Lt
Date: 09-2003
DOI: 10.1142/S0129167X03001995
Abstract: We show that if p:F→E is a covering of directed graphs, then the Cuntz–Krieger algebra C*(F) of F can be viewed as a crossed product of C*(E) by a coaction of a homogeneous space for the fundamental group π 1 (E). Combining this result with information about Cuntz–Krieger algebras gives some interesting corollaries which suggest conjectures about crossed products by coactions of homogeneous spaces of discrete groups. We then prove these conjectures.
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: Springer Science and Business Media LLC
Date: 10-2008
Publisher: Cambridge University Press (CUP)
Date: 23-08-2010
DOI: 10.1017/S0004972709001269
Abstract: 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 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.
Publisher: MDPI AG
Date: 11-07-2019
Abstract: A series of chloride receptors has been synthesized containing an amide hydrogen bonding site and a hydroquinone motif. It was anticipated that oxidation of the hydroquinone unit to quinone would greatly the diminish chloride binding affinity of these receptors. A conformational switch is promoted in the quinone form through the formation of an intramolecular hydrogen bond between the amide and the quinone carbonyl, which blocks the amide binding site. The reversibility of this oxidation process highlighted the potential of these systems for use as redox-switchable receptors. 1H-NMR binding studies confirmed stronger binding capabilities of the hydroquinone form compared to the quinone however, X-ray crystal structures of the free hydroquinone receptors revealed the presence of an analogous inhibiting intramolecular hydrogen bond in this state of the receptor. Binding studies also revealed interesting and contrasting trends in chloride affinity when comparing the two switch states, which is dictated by a secondary interaction in the binding mode between the amide carbonyl and the hydroquinone/quinone couple. Additionally, the electrochemical properties of the systems have been explored using cyclic voltammetry and it was observed that the reduction potential of the system was directly related to the expected strength of the internal hydrogen bond.
Publisher: World Scientific Pub Co Pte Lt
Date: 03-2014
DOI: 10.1142/S0129167X14500220
Abstract: We describe the primitive ideal space of the C*-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 ex les. 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.
Publisher: Cambridge University Press (CUP)
Date: 04-2004
Publisher: Elsevier BV
Date: 08-2017
Location: United Kingdom of Great Britain and Northern Ireland
Start Date: 2006
End Date: 2008
Funder: Australian Research Council
View Funded ActivityStart Date: 2015
End Date: 2017
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: 2009
End Date: 12-2012
Amount: $255,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2011
End Date: 12-2014
Amount: $300,000.00
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: 12-2018
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
View Funded ActivityStart Date: 2004
End Date: 12-2004
Amount: $20,000.00
Funder: Australian Research Council
View Funded Activity