Geometry of Banach Spaces and Operator Spaces

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Abstract for DMS-0200690 William B. Johnson and Gilles Pisier, PIs The principal investigators will investigate problems in the theories of Banach spaces and operator spaces. The problems to be considered fall into several subcategories; namely, affine approximation of Lipschitz functions, non linear infinite dimensional and finite dimensional geometry, finite dimensional subspaces of Lebesgue spaces, measures on infinite dimensional spaces, similarity theory, tensor products of operator spaces, bounded and completely bounded approximation properties, and amenability of spaces of operators. There will be coordination of the proposed project with the existing Workshop in Linear Analysis and Probability Theory at Texas A&M University. This Workshop encourages interactions among researchers and apprentices in different mathematical fields by bringing together graduate students, young investigators, and junior and senior postdoctoral participants in several areas of analysis. Banach space theory, the study of notions of "distance" which are not necessarily Euclidean on both finite and infinite dimensional vector spaces, forms the basis for both Euclidean and non-Euclidean geometry and the analytic geometry of infinite dimensional function spaces, including the space of continuous numerical functions on an interval as well as more complicated classes of functions. From the beginning, the ideas at the root of Quantum Mechanics have had a major impact on mathematics, stimulating the study of operators and their spectrum instead of numerical functions. These ideas have led recently to the new field of "Quantized Functional Analysis" or "Operator Space Theory" and this development has led to the solution of several major problems as well as opened new avenues of research. Analogously, the non linear geometry of Banach spaces also is a rather recent development which is beginning to have considerable impact in geometrical analysis in both the finite and infinite dimensional settings.