Operator algebras is a branch of mathematics that developed in the rigourisation of quantum mechanics. It primarily concerns two categories: C*-algebras and von Neumann algebras, and it involves interdisciplinary techniques within pure mathematics, drawing particularly on functional analysis, topology, and algebra.
Beyond their quantum mechanical origins, C*-algebra theory gained importance as it was quickly realised that C*-algebras can be constructed to capture key information about other mathematical objects. C*-algebras can be used to encode such things as symmetries, large data sets, networks, and dynamical systems. These constructions follow a common pattern: a mathematical object (such as a group of symmetries) is input and a C*-algebra is output. They suggest an important problem: what do properties of the output C*-algebra tell us about the input mathematical object?
Systematic analysis of this problem reveals that the key is in the classification of C*-algebras. Classification is motivated by the fact that different inputs may produce the same C*-algebra, although it can be difficult to see that they are actually the same. Classification solves this by proving that when C*-algebras agree on certain invariants, they are the same. Here, an invariant is a piece of information associated to a C*-algebra; the invariants used in the classification of C*-algebras are certain ones with many techniques available to compute them.
Recently, the problem of classifying C*-algebras has taken a spectacular turn, in which it has become apparent that certain C*-algebras cannot be classified with the traditional invariants, i.e., these invariants are not sensitive enough. Thorough examinations of why this is the case reveal that the obstructions to classification are related to high topological dimension.
The fact that topological dimension can be considered for C*-algebras is motivated by a classical theorem of Gelfand: that C*-algebras enjoying a special property called commutativity are simply encodings of topological spaces. No information is lost if one studies a space by looking only at this algebra of continuous functions. In particular, dimension of the space can be formulated in terms of properties of this algebra. Using the right formulation, dimension can then be generalised to noncommutative C*-algebras, providing each C*-algebra with a number, its dimension.
The study of C*-algebra dimension has revealed that:
(i) Finite dimension is a robust notion, equivalent to other properties (called regularity properties) for mysterious reasons, which is necessary for classification (and to some extent, sufficient);
(ii) For C*-algebra constructions, the particular value of dimension relates, in some cases, to known numerical invariants, while in others it produces a new interesting property.
These are observations largely at an empirical level (they hold for certain examples or special cases, suggesting that they should hold more generally); the basis of this project is to systematically investigate them, to show that they are true at deeper and more general levels. Point (i) is captured in a major conjecture of Toms and Winter, and we aim to prove this conjecture, and make the reasons less mysterious. For (ii), we aim to deepen our understanding of what C*-algebraic dimension means for C*-algebra constructions, making the relationships between dimension and other invariants more transparent.
Investigations into dimension thus far have opened up a number of new research directions, and this project will also pursue some directions that have potential for major impact.
These will:
- Produce new connections between the modern studies of C*-algebras and topology, and
- Provide new perspectives and inroads into the classification of C*-algebras.
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