Miss Becky Armstrong


Teaching and supervision



Thesis work

Thesis title: Simplicity of twisted C*-algebras of topological higher-rank graphs

Supervisors: Aidan SIMS, Nathan BROWNLOWE

Thesis 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 higher-rank 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 higher-rank graphs and topological graphs with the introduction of topological higher-rank 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" higher-rank graph C*-algebras using categorical 2-cocycles, and have developed a characterisation of simplicity of 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 higher-rank graphs. For each cofinal, proper, source-free topological higher-rank graph, and each continuous 2-cocycle on the associated path 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 étale groupoid C*-algebra 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 our desired characterisation of the simplicity of the twisted C*-algebra of a topological higher-rank graph. We show that the quotient of the path groupoid by the interior of its isotropy subgroupoid acts on the Cartesian product of the infinite-path 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 higher-rank 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.

To update your profile click here. For support on your academic profile contact .