Projects In Operator Algebra: Tensor Algebras, Coordinates, and Toeplitz Operators

University Of Iowa, Iowa City IA

Investigators

Abstract

ABSTRACT DMS-0070405 Muhly will study a variety of problems in operator algebra that may be divided into three groups: General Operator Algebras, Groupoids, and Toeplitz and Related Operators. Under the first heading are found a number of problems pertaining to the structure and representation theory of non-self-adjoint operator algebras. The theory of these algebras has been reinvigorated in recent years through advances in what has come to be known as quantized functional analysis, i.e., the theory of operator spaces. This subject coupled with the recent work of Muhly and Solel that identifies the C*-envelopes of various very general operator algebras allows one to make good progress on the program proposed more than thirty years ago by Wm. Arveson as a generalization of the Sz.-Nagy - Foias model theory of contraction operators on Hilbert space: study the representations of an operator algebra in terms of the C*-representations of its C*-envelope. In the area of groupoids, emphasis will be placed on the study of Fell bundles over groupoids. The impetus for this study comes from the representation theory of groupoids and from efforts to understand co-actions of groupoids. Further, these bundles (more accurately, generalizations of them) occur in the theory of product systems and the structure of non-self-adjoint operator algebras just mentioned. Toeplitz operators have long been a subject of intense interest in operator algebra/theory. In recent years Muhly and Xia have identified the automorphisms of the C*-algebra generated by all Toeplitz operators. The identification is described in terms of constructs that have played important roles in function theory and harmonic analysis on the disc. A particular importance is played by commutator singular integrals. Muhly proposes to extend these results, to the extent possible, to multivariable settings. Again, commutator singular integrals should play a decisive role, but the jump to dimensions greater than one presents difficult challenges. Muhly's work on non-self-adjoint operator algebras, while inspired primarily by developments in pure mathematics, has connections with problems of an applied nature, most particularly, mathematical systems theory - the theory of so-called H-infinity control. This, basically, is a body of knowledge dedicated to the problem of building machines, such as cars and airplanes, so that they meet certain performance criteria. Among these are such things as the responsiveness of the steering wheel, in the case of cars, or the rudder, in the case of airplanes. In short, Muhly's work will contribute to theoretical underpinnings of design problems ensuring the safety of all sorts of conveyances. Much of Muhly's work on tensor algebras over C*-correspondences, which has been inspired by the operator theory that underwrites this kind of control theory, seems to be usable in multivariable control problems. Indeed, quite serendipitously, it appears to be ready-made for multivariable problems. Muhly expects his research to provide a theoretical underpinning of multivariable control problems that here-to-fore have been handled by essentially ad hoc methods. Muhly's work in non-self-adjoint algebras and the theory of groupoids has made unexpected connections with quantum Markov processes - the theory of so-called open systems that arise in a variety of places such as the theory of lasers and materials science devoted to super conductivity. They also arise naturally in quantum field theory through the constructs known as sectors. It is anticipated that Muhly's work will help to provide the mathematical underpinnings of quantum field theory and advances in laser and materials science. Finally, Muhly's work on Toeplitz operators has made connections with perturbation theories that arise in areas of relativistic quantum theory and in quantum chemistry. It is anticipated that his projects on higher dimensional Toeplitz operators will have an increased impact in this arena.