Miss Becky Armstrong

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B_Armstrong

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.

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