Mathematical Sciences: Geometry of Banach Spaces and Operator Spaces

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9623260 W. B. Johnson and G. Pisier The proposers will investigate problems in the theories of Banach spaces and operator spaces. The problems to be considered involve: local theories of operator spaces and of spaces of measurable functions, polynomially bounded operators, bounded homomorphisms on C* algebras, infinite dimensional linear and nonlinear geometry of Banach spaces. Banach space theory forms the natural framework for studying geometries which are not Euclidean, while the theory of C* algebras is the standard framework for quantum mechanics. Operator spaces are the quantization (in the sense of quantum mechanics) of Banach spaces. This very new theory has already (e.g., in work of Junge and Pisier) had an impact on C* algebras, and part of this work involves approaching other classical problems in C* algebras/operator theory through operator spaces. The nonlinear geometry of Banach spaces is also in a neophyte stage, but work of Bourgain, Gorelik, Johnson-Lindenstrauss-Schechtman, and others suggest that this subject is ready to take-off. There is reason to hope that the nonlinear theory of Banach spaces will have applications in classical nonlinear analysis comparable to the many applications of the linear theory to classical linear analysis.