2004 IRISH SCIENTIST YEAR BOOK

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Queen's University Belfast
Martin Mathieu
What is Noncommutative Functional Analysis about?

Dr Martin Mathieu, leader of the research team NCFA

In Classical Physics, such as Electro- or Thermodynamics, measurements are mathematically represented by scalar-valued functions; e.g., temperature is assigned to a certain state of a system. In Quantum Physics, such as Quantum Mechanics or Quantum Field Theory, these functions are replaced by operators acting on an (infinite-dimensional) Hilbert space. The decisive difference in these two mathematical models lies in their algebra: while the product of two numbers a and b does not depend on the order, that is, ab=ba, and hence the same is true for the product of two functions, fundamental principles in Quantum Physics such as Heisenberg's Uncertainty Principle result in a noncommutative multiplication . If A and B are two operators on Hilbert space, then, typically, AB?BA!

Traditional Functional Analysis deals with functions and sets of functions; these are called function spaces or Banach spaces, after Stefan Banach who did ground-breaking work in this area in the 1930s. The new type of Functional Analysis deals with spaces of operators on Hilbert space. John von Neumann already pointed out at about the same time that, although these spaces can be studied as Banach spaces in an abstract way, this point of view will not be sufficient for their understanding. One has to allow for matrices of operators as well, and keep track of the so-called matricial structure of these spaces in order to fully understand the consequences of the noncommutative multiplication.

These operator spaces were studied in great detail since the 1970s, but only in 1988 a breakthrough was achieved by Ruan by characterising them in an abstract fashion very similar to the abstract characterisation of function spaces obtained by Banach. In the last 15 years, tremendous progress has been made by a large number of researchers � including Blecher, Effros, Junge, Paulsen, Pisier, and many others � in the understanding of the structure of operator spaces and, even more importantly, in their dynamical behaviour (that is time development of the corresponding physical system), which is described by completely bounded operators.

The Research Team in Noncommutative Functional Analysis (NCFA) at Queen's University Belfast at present consists of one Reader, two Lecturers, and three Research Students who are often visited by researchers and students from other institutions abroad. They have been researching into several aspects of the theory; for instance, in [1], a novel tool to study these completely bounded operators has been introduced and successfully applied to the solution of a number of problems.


Reference

[1] P. Ara, M. Mathieu: Local Multipliers of C*-Algebras, Springer-Verlag, London 2003.

Contact: Dr M Mathieu; E-mail: [email protected] ;Web: www.qub.ac.uk/mm