Mathematical Sciences: Complex Banach Spaces

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Banach space theory is a basic tool of mathematical analysis that has been of importance since its inception at the turn of the century. In recent years, there has been a rebirth of interaction with other aspects of mathematical analysis -- such as harmonic analysis and operator theory -- and probability theory. In particular, Banach space methods, in the hands of highly creative and powerful researchers, have led to fundamental breakthroughs in these other fields, and in turn probabilistic methods have enriched the structure theory of Banach spaces themselves. World centers of excellence in this area are in Israel, Paris, Warsaw, Ohio, and Texas. Professor Edgar's work reflects this broad point of view. The thrust of his work has been to make available over the complex field, the vast array of analytic methods that exist in the theory of real Banach spaces. His work is an interesting mixture of Banach space theory with complex analysis, probability (martingales), and harmonic analysis. In past work he has utilized pseudoconvex domains and plurisubharmonic functions to find analogs for complex Banach spaces of extreme points, vector measures, martingale convergence, and Choquet-type theorems. Professor Edgar has put into place a foundation on which to build a substantial theory. This subject, in which Professor Edgar is a leader, is in its infancy, and the current proposal will carry forward its development.