Operator algebras related to geometric structures

In the past 50 years, a major trend in Operator Theory focuses on the use of operator algebras for encoding geometrical and topological objects. Operator algebras may be considered as algebras of (bounded in norm) infinite matrices With complex entries. A central aspect of the program is to explore the passage from intrinsic properties of the object into properties of the associated operator algebras, and use invariants of the latter to classify the former. There are two interrelated questions that orient our study:

(q.1) Which (desirable) features of the object determine the operator algebra?
(q.2) What is the (desirable) level of equivalence for classifying objects?

Examples of examined objects so far include tilings, tangles, graphs, dynamical systems, groups, semigroups, varieties, homogeneous ideals, and stochastic matrices. The research so far incorporates operator algebras (both selfadjoint and non-selfadjoint) in terms of: representation theory, dilation theory, ideal structure, KMS-states theory, C*-envelopes, reflexivity, and hyperrigidity; for objects related to: C*-correspondences, product systems, subproduct systems, semigroup actions on operator algebras, and homogeneous ideals. For this project we will examine operator algebras arising from factorial languages.

It should be noted, that successful applicants will also be given the opportunity to complete teaching and demonstrating duties within the school amounting to up to £1500 per annum.