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

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ABSTRACTDMS-0070405Muhly will study a variety of problems in operator algebra that may bedivided into three groups: General Operator Algebras, Groupoids, andToeplitz and Related Operators. Under the first heading are found a numberof problems pertaining to the structure and representation theory ofnon-self-adjoint operator algebras. The theory of these algebras has beenreinvigorated in recent years through advances in what has come to be knownas quantized functional analysis, i.e., the theory of operator spaces. Thissubject coupled with the recent work of Muhly and Solel that identifies theC*-envelopes of various very general operator algebras allows one to makegood progress on the program proposed more than thirty years ago by Wm.Arveson as a generalization of the Sz.-Nagy - Foias model theory ofcontraction operators on Hilbert space: study the representations of anoperator algebra in terms of the C*-representations of its C*-envelope. Inthe area of groupoids, emphasis will be placed on the study of Fell bundlesover groupoids. The impetus for this study comes from the representationtheory of groupoids and from efforts to understand co-actions of groupoids.Further, these bundles (more accurately, generalizations of them) occur inthe theory of product systems and the structure of non-self-adjoint operatoralgebras just mentioned. Toeplitz operators have long been a subject ofintense interest in operator algebra/theory. In recent years Muhly and Xiahave identified the automorphisms of the C*-algebra generated by allToeplitz operators. The identification is described in terms of constructsthat have played important roles in function theory and harmonic analysis onthe disc. A particular importance is played by commutator singularintegrals. Muhly proposes to extend these results, to the extent possible,to multivariable settings. Again, commutator singular integrals should playa decisive role, but the jump to dimensions greater than one presentsdifficult challenges.Muhly's work on non-self-adjoint operator algebras, while inspired primarilyby developments in pure mathematics, has connections with problems of anapplied nature, most particularly, mathematical systems theory - the theoryof so-called H-infinity control. This, basically, is a body of knowledgededicated to the problem of building machines, such as cars and airplanes,so that they meet certain performance criteria. Among these are such thingsas the responsiveness of the steering wheel, in the case of cars, or therudder, in the case of airplanes. In short, Muhly's work will contribute totheoretical underpinnings of design problems ensuring the safety of allsorts of conveyances. Much of Muhly's work on tensor algebras overC*-correspondences, which has been inspired by the operator theory thatunderwrites this kind of control theory, seems to be usable in multivariablecontrol problems. Indeed, quite serendipitously, it appears to beready-made for multivariable problems. Muhly expects his research toprovide a theoretical underpinning of multivariable control problems thathere-to-fore have been handled by essentially ad hoc methods. Muhly's workin non-self-adjoint algebras and the theory of groupoids has made unexpectedconnections with quantum Markov processes - the theory of so-called opensystems that arise in a variety of places such as the theory of lasers andmaterials science devoted to super conductivity. They also arise naturallyin quantum field theory through the constructs known as sectors. It isanticipated that Muhly's work will help to provide the mathematicalunderpinnings of quantum field theory and advances in laser and materialsscience. Finally, Muhly's work on Toeplitz operators has made connectionswith perturbation theories that arise in areas of relativistic quantumtheory and in quantum chemistry. It is anticipated that his projects onhigher dimensional Toeplitz operators will have an increased impact in thisarena.