Mathematical Sciences: Problems in Operator Algebra

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Abstract Muhly Muhly will investigate a variety of problems in operator algebra that may be divided into three groups covered by the headings: General Operator Algebras, Groupoids, and Toeplitz and Related Operators. Under the first heading, he will pursue 5 strategic problems dedicated to understanding the structure of non-self-adjoint algebras. The focus will be on the categories of Hilbert modules associated with operator algebras. Inspiration for this derives from finite dimensional pure ring theory, but the key is to understand the interplay between the algebra and the metrics involved. In Muhly's work, groupoids provide concrete coordinates for representing operator algebras. The projects under this rubric are natural continuations of his earlier efforts. Most important among them, perhaps, is the problem of developing a good, topological cohomology theory for groupoids. The problems on Toeplitz operators that he proposes to investigate also derive naturally from his earlier work and progress on them should help shed light on important new connections between classical function theory/harmonic analysis and operator theory. Although in many respects, the detailed aspects of the problems Muhly will investigate are inspired by questions from core mathematics, their significance to applied mathematics and the "real world" is substantial. Much of Muhly's work contributes to the mathematical underpinning of control theory, H-infinity control theory, in particular, which, in turn, is the theoretical framework on which engineers base the design of aircraft, automobiles, robots, and other devices that need to respond to external stimuli and to be controlled to perform within specified guidelines. His work on groupoids and some of his work on Toeplitz operators is closely connected to issues of stochastic control, i.e., control problems where there are random uncertainties involved and one tries to steer a course that is best on the average. His work on operator algebras, generally, c ontributes to the field of multivariable control theory and to non-stationary control problems, i.e., problems where the control response is a function of the time of the input. Such problems are of increasing importance in the design of all types of devices, from acoustical filters, to digital processors, and neural networks.