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Topological C*-Categories

O'Sullivan, David Robert (2017) Topological C*-Categories. PhD thesis, University of Sheffield.

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Abstract

Tensor C*-categories are the result of work to recast the fundamental theory of operator algebras in the setting of category theory, in order to facilitate the study of higher-dimensional algebras that are expected to play an important role in a unified model of physics. Indeed, the application of category theory to mathematical physics is itself a highly active field of research. C*-categories are the analogue of C*-algebras in this context. They are defined as norm-closed self-adjoint subcategories of the category of Hilbert spaces and bounded linear operators between them. Much of the theory of C*-algebras and their invariants generalises to C*-categories. Often, when a C*-algebra is associated to a particular structure it is not completely natural because certain choices are involved in its definition. Using C*-categories instead can avoid such choices since the construction of the relevant C*-category amounts to choosing all suitable C*-algebras at once. In this thesis we introduce and study C*-categories for which the set of objects carries topological data, extending the present body of work, which exclusively considers C*-categories with discrete object sets. We provide a construction of K-theory for topological C*-categories, which will have applications in widening the scope of the Baum-Connes conjecture, in index theory, and in geometric quantisation. As examples of such applications, we construct the C*-categories of topological groupoids, extending the familiar constructions of Renault.

Item Type: Thesis (PhD)
Keywords:
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Mr David Robert O'Sullivan
Date Deposited: 13 Apr 2017 13:11
Last Modified: 13 Apr 2017 13:11
URI: http://etheses.whiterose.ac.uk/id/eprint/16775

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