Teaching and supervision
Timetable
B_ArmstrongThesis work
Thesis title: Simplicity of twisted C*algebras of topological higherrank graphs
Supervisors: Aidan SIMS, Nathan BROWNLOWEThesis abstract:
The theory of directed graph C*algebras emerged in the late 1990s and has since developed into a major area of research within the field of operator algebras, with applications to noncommutative geometry and the classification of simple C*algebras.
Directed graphs and their C*algebras have been generalised in a number of ways. In 2000, Kumjian and Pask introduced the notion of a higherrank graph and its associated C*algebra. In 2004, Katsura took the generalisation of directed graph algebras in a different direction by constructing a C*algebra from a topological graph. In 2005, Yeend unified the notions of higherrank graphs and topological graphs with the introduction of topological higherrank graphs. He employed groupoid techniques to construct C*algebras associated to these graphs, generalising the C*algebras of Kumjian–Pask and Katsura. More recently still, Kumjian, Pask, and Sims have investigated the effect of "twisting" higherrank graph C*algebras using categorical 2cocycles, and have developed a characterisation of simplicity for these algebras in terms of the underlying graphical and cohomological data.
In this thesis, we initiate the study of twisted C*algebras associated to topological higherrank graphs. For each cofinal, proper, sourcefree topological higherrank graph, and each continuous 2cocycle on the associated boundarypath groupoid, we construct a twisted groupoid C*algebra in the sense of Renault. By extending results of Brown, Nagy, Reznikoff, Sims, and Williams, we characterise the injectivity of homomorphisms of a twisted C*algebra associated to a Hausdorff étale groupoid in terms of the injectivity of homomorphisms of the twisted C*algebra associated to the interior of the isotropy. We apply this result to establish the desired characterisation of simplicity of the twisted C*algebra of a topological higherrank graph. We show that the quotient of the boundarypath groupoid by the interior of its isotropy subgroupoid acts on the Cartesian product of the infinitepath space of the graph and the dual group of a particular subgroup of the periodicity group of the graph that is dependent on the cohomological data. We then prove that the twisted topological higherrank graph C*algebra is simple if and only if this action is minimal. Our characterisation of simplicity generalises the analogous result of Kumjian, Pask, and Sims.
Selected grants
2018
 Selfsimilar actions and their C*algebras; Whittaker M, Armstrong B; Office of Global Engagement/Travel Grants.
Selected publications



