This research can be divided into three themes.
Michael Dritschel has worked together with McCullough and Woerdeman on concrete aspects of an abstract model theory developed by Agler, and with McCullough has made an important contribution to the long-standing rational dilation problem. He's also worked on generalisations of the famous Nevalinna–Pick problem, which have applications in control engineering. Together with Vinnikov he is working to develop a real function theory for functions in non-commuting variables.
The spectrum of operators has surprising holomorphic and subharmonic properties. These can used to study the spectrum preserving maps between algebras of operators, in particular to study the Kaplansky conjecture, that the Jordan algebra structure is determined by the spectral structure of an algebra.
Ransford and White found a spectral characterization of algebraic operators. Their results have recently been applied to the study of spectrum preserving maps between algebras (such as von Neumann algebras) which have a large number of algebraic elements.
More recently White has been using the approach of several complex variable to solve questions about structured interpolation problems, where one seeks to find controllers for linear systems which are only known up to small structured perturbations in the plant. This control problem, as usual, is solved by reducing it to a generalized Nevanlinna–Pick problem. White's approach, based on several complex variables, complements the Dritschel's approach, which is based on operator theory.
Many naturally arising algebras come with a topology: for example, the L1-algebra of a locally compact group and the algebra of infinitely differentiable functions on a manifold are a Banach space and a Fréchet space respectively. Newcastle is one of the homes of the homology theory of topological algebras, and Zina Lykova and Michael White continue the tradition. Both are especially concerned with the calculation of continuous Hochschild and cyclic homology and cohomology and with general results that facilitate such calculations. White, with others, has calculated the cohomology of numerous L1-algebras. For example, he with Gourdeau and Johnson has shown that l1(Z+) has trivial simplicial cohomology in dimensions above 1. Gourdeau, Lykova and White have proved a version of the Künneth formula for the continuous cohomology of certain projective tensor product algebras and thereby obtained explicit descriptions of the continuous simplicial cohomology of l1(Zk+) and L1(Rk+). Lykova is particularly interested in the cohomological problems that arise in non-commutative geometry. For this application one needs to calculate the Hochschild and cyclic cohomology of Fréchet and more general topological algebras, to which end familiar homological techniques (excision, use of projective and inductive limits, Künneth formulae) need to be extended to the locally convex setting. Lykova is in the course of developing an extensive theory for this purpose.