The area of research within Pure Mathematics I have been involved in over the last number of years can be most adequately described as Noncommutative Functional Analysis.

Traditional Functional Analysis, as it was formed by Stefan Banach early in the 20th century on the basis of the work of many others, seeks to understand Banach spaces and operators between them. A classical Banach space can be viewed as a space of functions.
Noncommutative Functional Analysis grew out of the theory of C* -algebras, which are algebras of bounded linear operators on Hilbert space. The new structure arises by enriching a Banach space with a 'noncommutative' structure allowing an embedding into a space of operators on Hilbert space. These operator spaces were axiomatically characterised by Zhong-Jin Ruan in 1988 and their main new feature is the additional level of matrices with operator entries (matrix-normed spaces). An overview on the theory can be found here.

Being a subspace of an algebra (the algebra of all bounded linear operators on a Hilbert space) whose multiplication is in general noncommutative, many new features appear for an operator space which are invisible or non-existent at the Banach space level. This makes the theory significant and useful for many modern applications, e.g., in quantum information theory.

Naturally, algebraic methods become very important. In longstanding collaboration with Prof. Pere Ara (Barcelona) we developed a new tool called the local multiplier algebra of a C*-algebra, and we published a research monograph on this topic. This concept enables us, for example, to solve many more equations occurring in the study of operators between C *-algebras. The construction parallels the so-called symmetric ring of quotients in Noncommutative Ring Theory.

A generic question is of the form:

• What is the structure of a (bounded, linear) operator on a given C*-algebra compatible with some of the given structure?

Depending on the structure we have in mind we are led to: Lie derivations and Lie homomorphisms; commutativity-preserving mappings; completely bounded and completely positive operators; spectrally bounded operators and spectral isometries; and many others.

The latter class is a prime example of a situation that cannot be understood from the commutative viewpoint.

Recent introductions to, and surveys on, my work can be found

• on local multipliers and more general C* -algebras of quotients, here;

• on spectral isometries and their relation to an old problem by Kaplansky, here.

I no longer offer supervision of new PhD students.

Supervision of previous Research Students:

So far I supervised 7 Diploma (Master's) Theses, 1 MPhil Thesis at QUB, and 6 Doctoral (PhD) Theses at the University of Tübingen (Germany) and at QUB.
A list of the titles is given below.

1. Diploma (Master's) Theses

The structure of the generators of norm continuous semigroups of completely positive operators on C*- algebras, February 1992.

Applications of completely bounded mappings in the theory of Hilbert space operators, May 1993.

Foundations of a noncommutative topology, September 1993.

Inner derivations of C*- algebras, November 1994.

Examples of noncommutative geometries, March 1995.

News on the unbounded Kleinecke-Shirokov conjecture, November 1998.

Homology theory for commutative C*- algebras, August 1999.

The norm problem for elementary operators on C*- algebras, October 2004.


2. Doctoral (PhD) Theses

Jürgen Schweizer, Interplay between noncommutative topology and operators on C*- algebras, February 1997.

Ralf Banning, Commutativity preserving mappings on C*- algebras, April 1998.

Gerhard Schick, Spectrally bounded operators on Banach algebras, December 2001.

Conal Ruddy, The structure of spectral isometries on C*- algebras, December 2006.

Matthew Young, The structure of spectrally bounded operators on Banach algebras, March 2016.

Michael Rosbotham, Cohomological dimension for C*-algebras, September 2021.

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