My research lies in Operator Algebra Theory, a branch of Functional Analysis – an area that focuses on the study of operators acting on Hilbert spaces. It blends together algebraic and topological features into a beautiful mathematical subject with an increasing number of applications to other fields.
Historically, there have been two main sources of motivation for the development of the field. The first one comes from Quantum Mechanics. Murray and von Neumann defined in the 1930’s what is known today as von Neumann algebras – special collections of bounded linear operators, which model complex quantum systems. If one needs to single out one most important feature that make von Neumann algebras useful to describe quantum, as opposed to classical, phenomena, this would be non-commutativity: a von Neumann algebra is a structure, whose elements can be multiplied, but the commutativity rule ab = ba does not hold true in general. In the 1940’s, Gelfand and Naimark introduced structures, known as C*-algebras, that were more general than von Neumann algebras and were the appropriate context to study non-commutative topological phenomena.
Again in the 1930’s, Banach formulated the simple axioms of what is known today as a Banach space, which appeared as a unifying abstract concept containing as special cases an enormous number of important examples of linear spaces that had already been studied for the purposes of Analysis.
It the 1980’s the two streams were blended together through Operator Space Theory. This quickly developing field has penetrated various branches of Analysis and has found applications to other scientific areas such as Quantum Computing. Part of my research lies in this discipline and is focussed on objects called operator systems – spaces of bounded linear operators acting on Hilbert space, containing the identity operator and closed under taking the adjoint. In recent joint work with A. Kavruk, V. Paulsen and M. Tomforde, we studied tensor products in the operator system category as well as properties of operator systems expressed in terms of tensor products. A number of questions and directions remain to be explored, including intriguing connections with Graph Theory.Interactions between Operator Algebras and Harmonic Analysis
Another direction in my current research is identifying various links between Operator Algebras and Abstract Harmonic Analysis. The latter area focusses around (generally, non-commutative) topological groups and their representations. There is a fruitful flow of ideas in both directions and a number of problems to be tackled. One example are the concepts of Schur and Herz-Schur multipliers – these are functions which give rise to certain transformations that act on C*-algebras, having a number of applications within Operator Theory and Harmonic Analysis.Invariant Subspace Theory
Yet a third circle of problems I am working on is Invariant Subspace Theory. A naturally important question is to determine the invariant subspaces of a given collection of operators. Such collections of subspaces are called reflexive (or invariant) subspace lattices. Tensor products can also be used here to form more complex, but still tractable, subspace lattices, from given ones. Tensor product formulas relate the invariant subspace lattice of the tensor product of two operator algebras to the tensor product of the corresponding invariant subspace lattices. When does such a formula hold? This question is vastly open, although some partial results, for particular situations, are known.
I am happy to provide supervision to students who wish to pursue a PhD or an MPhil degree in the above areas. Below you may find a list of my former and current PhD students, as well as some references, which give a more concrete idea of the problems I work on.
Joe Habgood, Convergence properties of bimodules over maximal abelian selfadjoint algebras, 2007.
Martin McGarvey, Normalisers of reflexive operator algebras, 2009.
Savvas Papapanaiydes, Properties of subspace lattices related to reflexivity, 2011.
Naomi Steen, Closable multipliers, in progress.
Tensor products of operator systems, J. Funct. Anal. 261 (2011), 267-299 (with A.Kavruk, V. I. Paulsen and M. Tomforde)
Operator algebras from the Heisenberg semigroup, Proc. Edinburgh Math. Soc. 55 (2012) vol. 1, 1-22 (with M. Anoussis and A. Katavolos)
Closable multipliers, Int. Eq. Operator Th. 69 (2011), vol. 1, 29-62 (with V. S. Shulman and L. Turowska)
Schur and operator multipliers, Banach Centre Publications 91 (2010), 385-410 (with L. Turowska)