W*-BUNDLE TECHNIQUES AND THE STRUCTURE OF SIMPLE C*-ALGEBRAS

"C*-algebras are mathematical objects that arose from the rigourisation of quantum mechanics. Each C*-algebra is a set of continuous linear maps from a Hilbert space to itself, closed under a few natural algebraic and analytic operations. Upon their inception, it was quickly realised that C*-algebras can be created in canonical ways from many other mathematical objects, modelling such things as symmetries, time-evolving systems, and large data sets. Time and again, interesting relationships have manifested between properties of the object being input and those of the resulting C*-algebra.

For some time, it has been quite clear that different constructions can produce the same C*-algebra; this is interesting externally, where it may imply a profound relationship between the differing input data, and internally, where it allows single C*-algebras to be studied using the different techniques available from each different way of constructing it. However, a thorough elucidation of what conditions on the input objects produce different C*-algebra outputs has yet to be achieved. Achieving this goal amounts to classifying C*-algebras: showing that suitable, computable invariants (primarily, K-theory) are sufficiently sensitive to always distinguish different C*-algebras.

It has recently become apparent that to classify C*-algebras, one should study regularity properties of the C*-algebras - certain properties of C*-algebras that indicate they are less complex and more tractable. Regular C*-algebras are ones that have low (topological) dimension - in a way that exactly generalises dimension of a space. Just as low dimensional spaces are easier to visualise, it is often easier to prove things about them, to the extent that certain things that are true of all low-dimensional spaces are no longer true in higher dimensions. This carries forward to C*-algebras: more and better things can be proven about low dimensional C*-algebras than high dimensional ones. Returning to classification, it has been shown in many cases that C*-algebras whose invariants take the same value are automatically the same (or isomorphic), provided that the C*-algebras have low dimension.

I have been involved in research concerning regularity, and have found that a certain recent tool called W*-bundles shows tremendous promise, although its fundamental theory has yet to be developed. From a C*-algebra, one produces a W*-bundle, and uses this as a tool.
This works because:
(i) the W*-bundle has more structure, and it seems that it should be easier to prove things about it than about the C*-algebra;
(ii) the W*-bundle has a very special relationship to the C*-algebra - it contains it in a special way - so that facts about the W*-bundle can have important implications for the C*-algebra.

The aim of this project is to further our understanding of structure and classification of C*-algebras, by developing the theory of W*-bundles."