Operator Algebra: David’s primary interest is in a field of Pure Mathematics called Operator Algebra. Operator algebras were first formulated in the 1930’s as Mathematicians began to formalise quantum mechanics. Since then operator algebras have played a role in many areas of Mathematics and theoretical Physics. Specifically, David is interested in the operator algebras generated by operators which satisfy relations encoded by a directed graph. His work has shown that many properties of the operator algebra can be read off from properties of the directed graph. There are also interesting connections with combinatorial topology and nonabelian duality. Lately, I have been working on higher dimensional versions of directed graphs, called k-graphs, which contain much more information and provide the potential for new applications Field of Study: Functional analysis, operator algebra, nonabelian duality, symbolic dynamics Professional Activities and Affiliations: • Member, Institute for Mathematics & Its Applications • Member Australian and American Mathematical societies • NSF-CBMS Regional Conference in the Mathematical Sciences: ‘Graph algebras: Operator Algebras We Can See’, University of Iowa (2004) • CI with Aidan Sims, ‘Pictures for operator algebras: higher-rank graphs’, ARC Discovery—Projects grant (2006-2009) • CI with Eilers, Elliott, Kumjian, Raeburn and Toms. BIRS workshop 08w5034, 'C*-algebras associated to discrete and dynamical structures ' (2008) • CI with Aidan Sims, 'Oeprator algebras associated to groupoids', ARC Discovery—Projects grant (2010-2011) • Sole CI, 'Higher dimensional methods for algebras and dynamical systems', ARC Discovery—Projects grant (2011-2013) • CI with Aidan Sims, ‘Cohomology, symbolic dynamics and operator algebras’, ARC Discovery—Projects grant (2012-2104) • Director 55th Annual meeting of Australian Mathematical Society meeting, Wollogong 26-29 September 2011 Current Research Students: • Benjamin Maloney : Semigroup actions on higher-rank graphs and their graph C*-algebras • Hui Li : Topological graphs, Hilbert modules and C*-algebras • Yuxiang Tang : Graphical Approach to C*-algebras
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A family of 2-graphs arising from two-dimensional subshifts (with Iain Raeburn and Natasha A. Weaver), Faculty of Informatics - Papers (2009)
Higher-rank graphs (or k-graphs) were introduced by Kumjian and Pask to provide combinatorial models for...
Crossed products of k-graph C*-algebras by Zl (with Cyntyhia Farthing and Aidan Sims), Faculty of Informatics - Papers (2009)
An action of Zl by automorphisms of a k-graph induces an action of Zl by...
Noncommutative manifolds from Graph and k-Graph C*-algebras (with Adam Rennie and Aidan Sims), Communications in Mathematical Physics (2009)
In [PRen] we constructed smooth (1,∞)-summable semifinite spectral triples for graph algebras with a faithful...
C*-algebras of labeled graphs (with Teresa G. Bates), Journal of Operator Theory (2007)
We describe a class of C*-algebras which simultaneously generalise the ultragraph algebras of Tomforde and...
Rank-two graphs whose C*-algebras are direct limits of circle algebras (with Iain Raeburn, Mikael Rordam, and Aidan Sims), Journal of Functional Analysis (2006)
We describe a class of rank-2 graphs whose C* -algebras are AT algebras. For a...