Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. Thus, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum theory, etc. could be studied productively via the integral equations that model the phenomena. So it has been, that from the fertile minds of Hilbert, von Neumann, and other giants that the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. Also central to Modern Analysis is the related discipline of operator algebras in which one studies collections of operators simultaneously. As the mathematical construct which best transfers the concepts of probability, measure theory, topology, and more recently geometry to noncommutative contexts, operator algebras relate to a rich, effective, and important array of applications. Professors Bunce, Paschke, Salinas, and Upmeier are all leaders in the areas of operator theory and operator algebras, and their research group at Kansas is one of the strongest and most versatile at any university in the United States. Collectively, this group covers a broad span of these disciplines. Professor Bunce has been a major contributor to the C*-algebra version of the Stone-Weierstrass theorem, to abstract operator theory proper, and to applications in systems theory. Professor Salinas has made important contributions to C*-algebraic K-theory applied to single operator theory and several complex variables. Professor Paschke has made fundamental contributions to the study of concrete C*-algebras in the context of noncommutative algebraic topology. Professor Upmeier has a deep and penetrating understanding of bounded symmetric domains. He has been a major figure in the study of Toeplitz and Hankel operators on these domains. In the current proposal, Bunce will study dilation problems in operator theory, Bishop's theorem, and operator means and inequalities. Paschke will work on C*-algebras associated with discrete groups and will pursue problems dealing with irrational rotation C*-algebras. Salinas will continue his research on the joint behavior of n-tuples of operaors, gaining insight into the interplay between the spectral and C*-algebraic properties of the canonical n-tuple for classes of kernel functions. Upmeier will investigate a symbol calculus and index theory for multivariable Toeplitz operaors on pseudoconvex domains in complex n-space. He will also study non-type I Toeplitz C*-algebras related to automorphic functions and quantum mechanics. There are three postdoctoral mathematicians working with this group in a mentorial relationship. Their interests spread across Lie group C*-algebras, symplectic geometry, singular foliations, holomorphic curves in the Grassmanian, and noncommutative integration.
