Research

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.

Historical overview

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.

Some research directions

Operator Systems and Non-commutative Graphs

It the 1980’s the two streams were blended together through Operator Space Theory. This quickly developing field has penetrated a number of 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 a joint work with D. Farenick, A. Kavruk, V. Paulsen and M. Tomforde, we studied tensor products and quotients in the operator system category as well as properties of operator systems expressed in terms of these constructs. Recently, it has become apparent through the work of R. Duan, S. Severini and A. Winter that better understanding of properties of operator systems is important for progress in aspects of Quantum Information Theory, and more specifically on questions regarding quantum compression rates, zero error capacities of quantum channels and quantum simulation. In that respect, operator systems play the role of confusability graphs introduced by Shannon and used to express the zero-error capacity of a classical information channel. Intriguing connections between operator systems and graphs suggest ways to lift combinatorial problems to the realm of Operator Algebra Theory, giving rise to a plethora of questions and directions for investigation.

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, pioneered by W. Arveson, P. Eymard and T. Varopoulos, which has led to the formulation of a research programme that aims at identifying suitable operator theoretic versions of concepts and results from Harmonic Analysis. 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. Another example is the notion of sets of multiplicity, which has its roots in Cantor's study of uniqueness of Fourier series representations. The main research prospect of this programme is the possibility of employing operator theoretic methods in harmonic-analytic investigations.

Invariant Subspace Theory and Non-selfadjoint Operator Algebras

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. This research direction is related to the introduction and study of new classes of non-selfadjoint operator algebras and is linked to objects of not necessarily operator theoretic nature, such as dynamical systems. One of the main tools that is used to tackle specific problems is the coordinate representation of classes of non-selfadjoint algebras, a versatile area employing a number of concepts such as groupoids, graphs and semi-groups.

Supervision

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.

PhD students:

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, Unbounded generalisations of operator multipliers, 2013.

Linda Mawhinney, Inductive limits of operator systems, 2016

Andrew McKee, Multipliers of dynamical systems, 2017

Weijiao He, Interactions between topological groups and operator algebras, in progress

Gareth Boreland, Continuous graphs in quantum information theory, in progress

Some relevant references:


Estimating quantum chromatic numbers, J. Funct. Anal. 270 (2016), 2188-2222 (with S. Severini, D. Stahlke, V. I. Paulsen and A. Winter)

Sets of multiplicity and closable multipliers on group algebras, J. Funt. Anal. 268 (2015), 1454-1508 (with V. S. Shulman and L. Turiwska)

Operator systems from discrete groups, Comm. Math. Phys. 329 (2014), 207-238 (with D. Farenick, A. Kavruk and V. I. Paulsen)

Quotients, exactness and nuclearity in the operator system category, Adv. Math. 235 (2013), 321-360 (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)

Tensor products of operator systems, J. Funct. Anal. 261 (2011), 267-299 (with A. Kavruk, V. I. Paulsen and M. Tomforde)

Closable multipliers, Integral 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)

Herz-Schur multipliers of dynamical systems, Adv. Math.  331 (2018), 387-438 (with A. McKee and L. Turowska)

Complexity and capacity bounds for quantum channels, arXiv:1710.06456, to appear in IEEE Trans. Inf. Theory (with R. Levene and V. I. Paulsen)