Operator Algebras and Operator Theory

In the 1940's Murray and von Neumann developed the mathematical framework underlying Quantum Mechanics by means of Hilbert space operator theory and the analysis of "Rings of Operators". In contemporary times operator theory plays an important role in diverse areas of mathematics and its applications, including, for example, the emerging mathematical methodology for Quantum computers. In their seminal papers Murray and von Neumann identified the need to classify the "simple" algebras (those with trivial centre) and the topic of classification remains a central one in the modern theory.

Research at Lancaster has made diverse contributions to the classification and structure of various antisymmetric operator algebras, in particular, to limit algebras, to translation-dilation algebras, and to operator algebras whose generators are associated with finite and infinite directed graphs, the so called free semigroupoid algebras.

Free semigroup and free group algebras also have significant interplay with models in the theory of random matrices.

About the Operator Theory Group.

The group consists of Professor S.C. Power, Dr. G. Blower, Dr. Niels Laustsen, and Professor Martin Lindsay .

Additionally the group benefits from international collaborations with Professor Keneth Davidson (Waterloo), Professor D. Kribs (Guelph, an EPSRC VF in 2006), Professor A. Katavolos (Athens), Professor Baruch Solel (an EPSRC VF in 2006), and Professor Alan Hopenwasser (Alabama).

The group ran its third London Mathematical Society International Research Workshop at Ambleside in July 2004, Operator Algebras and Random Matrices. The workshop concerned advances in

• free semigroup algebras;
• free probability, information and entropy;
• the mean field approach to random matrix theory.

Some key publications:

• A. Katavolos and S.C. Power, Translation and dilation invariant subspaces of L2(R), Journal für die reine und angewandte Mathematik (Crelle's Journal), 552 (2002), 101-129. Crelle
• S. C. Power, Approximately finitely acting operator algebras, J. Functional Anal., 189 (2002), 409-468. JFA arXiv
• D. W. Kribs and S.C. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc., 19 (2004), 117-159. arXiv
• G. Blower, Almost sure weak convergence for the generalized orthogonal ensemble, Journal of Statistical Physics, 105 (2001), 309-335. JSP
• N. J. Laustsen and R.J. Loy, Closed ideals in the Banach algebra of operators on a Banach space. Topological algebras, their applications, and related topics, 245--264, Banach Center Publ., 67, Polish Acad. Sci., Warsaw, 2005.

Research Supervision:

The current research interests of Professor Stephen Power are mainly in nonseladjoint operator algebra. He is prepared to supervise PhD research projects in the following areas:

Limit Algebras.

These are operator algebras which are determined by a chain of finite dimensional algebras. One seeks to understand their isomorphism type and their structure by some kind of generalised finite dimensional analysis. The basic theory of these algebras is summarised in my book. Also my recent Lisbon summer school lectures give an introduction to these and to more general subalgebras of graph C*-algebras. See [1],[2],[3].

Reflexive Algebras, Lie Semigroups and Subspace Manifolds

Aristides Katavolos and I showed in 1997 that the classical Weyl commutation relations $U_tV_s = e^{its}V_sU_t$ give rise to a novel reflexive operator algebra whose invariant subspaces form a 2 dimensional manifold, naturally parametrised as a quarter sphere. We subsequently began a general theory of Lie semigroup algebras which presents many fascinating problems linking Lie semigroups and subspace manifolds. This is a current area of research with past student Rupert Levene at Belfast. An introductory account is given in my Seville lectures in 2004. See [4],[5],[6]

Operator Algebras, Graphs and Higher Rank Graphs.

To each directed graph G one can associate a Fock space Hilbert space, whose basis is indexed by all the finite paths of the graph, together with natural shift operators (partial creation operators). David Kribs and I developed the theory of the weakly closed nonselfadjoint algebras $L_G$ generated these operators. The algebras may be finite dimensional, they may have function matrix representations, or, in contrast, they may possess subalgebras which are freely noncommutative. In particular these algebras generalise the motivating examples of free semigroup algebras studied by Davidson and Pitts (and which correspond to G with a single vertex). Current collaborative research is aimed at extending the theory to the fashionable context of higher rank graphs. See [7],[8] below.

Geometric Rigidity Theory and Rigidity Operators.

In recent joint work I have contributed to the analysis of constraint systems and the equation solving that underlies CAD (Computer Aided Design) software, on the one hand, and the analysis of rigidity theory for bar-joint frameworks on the other. In current work the development of a theory of flexibility and rigidity for infinite bar-joint frameworks has begun. This is an interesting topic from a pure mathematical perspective benefitting from techniques in operator theory and functional analysis. Furthermore, part of the motivation for such a study derives from the mathematical models used in the analysis of repetitive structures in engineering, and the analysis of amorphous and periodic structures in materials science.

The current techniques required of Rigidity Theory are already somewhat hybrid, drawing on methods in graph theory, combinatorics, algebraic geometry and analysis. This, together with the novelty of infinite rigidity frameworks, makes it a very suitable topic for PhD projects.

1. S.~C. Power, {\em Limit algebras: an introduction to subalgebras of {$C\sp *$}-algebras}, Pitman Research Notes in Mathematics Series, vol. 278, (Longman Scientific \& Technical, Harlow, 1992) CRC Press ISBN: 0582087813.
2. S.~C. Power, Approximately finitely acting operator algebras, J. Functional Anal., 189 (2002), 409-468.
3. S.~C. Power, Subalgebras of graph C*-algebras, Lectures for the Summer School Course given at the ICM 2006 satellite workshop on Operator Algebras, Operator Theory and Applications, held in Lisbon, 1-5 September 2006. To appear in the conference proceedings published in the Birkhauser book series, Operator Theory: Advances and Applications.
4. A. Katavolos and S.C. Power, Translation and dilation invariant subspaces of $L^2(R)$, Journal für die reine und angewandte Mathematik (Crelle's Journal), 552 (2002), 101-129.
5. R.L. Levene and S.C. Power, Reflexivity of the translation-dilation algebras on $L^2(\mathbb{R}$, International J. Math., Vol 14 No 10 (2003), 1081-1090
6. S.C.Power, Invariant subspaces of translation semigroups, Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Ed. Afonso Montes Rodr\'iguez, University of Seville, 2006.
7. S.C. Power and D. Kribs, Free semigroupoid algebras, J. Ramanujan Math. Soc., 19 (2004), 117-159.
8. S.C. Power and D. Kribs, The $H^\infty$ algebras of higher rank graphs, preprint, August 2004, Math. Proc.of the Royal Irish Acad., 106 (2006), 199-218.
9. K. R. Davidson, S.C. Power and D. Yang, Atomic representations of rank 2 graph algebras, Journal of Functional Analysis, Volume 255, (2008), 819-853.
10. J.C. Owen and S.C. Power, The non-solvability by radicals of generic 3-connected planar Laman graphs, Trans. Amer. Math. Soc., 359 (2007), 2269-2303.
11. J.C. Owen and S.C. Power, Infinite bar-joint frameworks, Proceedings of the Symposium in Applied Computing, Honolulu, Hawaii, 2009.
12. J.C. Owen and S.C. Power, Frameworks, symmetry and rigidity, arXiv:0812.3785.