This website gives some details of the research currently being undertaken in the Mathematics Department of University College Cork, Ireland in the area of operator algebras and related areas, such as single operator theory, quantum probability and the C*-algebra approach to quantum groups. Information is given on some current projects and on the research staff and postgraduate students working in these areas. Some of the staff maintain personal webpages and further information can be found at their websites.

The following is a list of lecturing staff and postgraduate students in the Operators Algebras Group:

Adel Badi (PhD student of Prof. Gerard J Murphy)

e-mail: A.Badi@ucc.ie

*** (Postdoctoral assistant of Prof. Gerard J Murphy) This position is currently vacant and is in process of being advertised and filled. Recent holders of this post have been Drs Johan Kustermans, Thomas Hadfield and Lars Tuset.

Gerard J. Murphy (Associate Professor of Mathematics)

e-mail: gjm@ucc.ie

Stephen Wills (Lecturer)

e-mail: S.Wills@ucc.ie

The UCC Operator Algebras Group belongs to the European Union Operator Algebras Network. This is devoted to research in quantum spaces and noncommutative geometry. The Network group in Ireland also incudes some members of staff of the Dublin Institute of Advanced Studies. The Irish coordinator for the Network is Prof. Gerard J Murphy.

Listed below are some current research projects of the staff in the UCC Operator Algebras Group.

Generalized Toeplitz operators and algebras (Gerard Murphy and Adel Badi) Generalized Hardy spaces occur in a variety of areas. One extremely large class of such spaces was introduced in the 1960s as an appropriate abstract setting in which to study function theory and as a tool in the analysis of function algebras. Another large class arises from the study of bounded symmetric domains in several complex variables theory. Associated to all of these generalized Hardy spaces are corresponding Toeplitz operators. These have many properties in common with the usual class of Toeplitz operators on the circle but there are also profound differences. For instance, in general the index theory for these operators is real-valued, not integer-valued. For this and other reasons it is appropriate and convenient to study these operators by means of the C*-algebras that they generate. These Toeplitz algebras have many properties that make them interesting and important. At present we are concerned with continuing an earlier analysis of these algebras and with extending the index theory, which to date is known only for certain classes of Toeplitz operators.

Quantum groups (Gerard Murphy) A unital C*-algebra with a comultiplication is the quantum analogue of a compact semigroup. If certain cancellation properties are also assumed, one has a quantum analogue of a compact group, called a compact quantum group. One can also define algebraic quantum groups and locally compact quantum groups. Together with my co-workers Eric Bedos and Lars Tuset in Oslo, Norway, I have investigated conditions for amenability and co-amenability of quantum groups.

Noncommutative geometry (Gerard Murphy) In a series of papers Gerard Murphy, Johan Kustermans (Leuven, Belgium) and Lars Tuset (Oslo, Norway) studied differential calculi over quantum groups. The analysis revealed that the volume integrals in this setting are not graded traces in the sense used in Alain Connes's version of noncommutative differential geometry, but rather that they have a property similar to that of a KMS-state on a C*-algebra. Thus, the theory of quantum differential calculi is in some sense a "Type III" theory and it may be that it does not fit into Connes framework without further modification of that theory. We showed that a twisted cyclic cohomology is the appropriate tool to use in this setting and obtained a large number of results in this area, including a Hodge-type decomposition and a version of Poincare duality. Gerard Murphy is now exploring Chern character theory in the setting of twisted cyclic cohomology.

Quantum probability (Steve Wills) My research is in quantum or noncommutative probability, a subject that combines mathematical physics, probability theory and functional analysis. I generally focus on the third one of these, in particular using aspects of operator theory, the theory of operator algebras and spaces, and semigroup theory to obtain existence results for quantum stochastic differential equations, as well as studying the properties of their solutions. I am interested in possible applications of this work not only to real physical models, but also in noncommutative geometry, as well as to the dilation theory of quantum dynamical semigroups and their interaction with E_0-semigroups.

Other Staff: There are other members of staff of the School of Mathematics in UCC who have interests related to operator algebras. Principally, one should mention Prof. Neil O'Connell, part of whose research lies in the area of random matrices.