The Isomorphic Structure of Banach and Operator Spaces

University Of Texas At Austin, Austin TX

Investigators

Abstract

ABSTRACT: This proposal deals with problems concerning the isomorphic structure of classical Banach spaces and their quantized analogues, operator spaces. The problems to be studied lie in the following four areas: I. The complementation problem for ideals in C* algebras. II. Certain extension properties for separable operator spaces. III. The structure of complemented subspaces of nuclear C*-algebras. IV. The structure of non-commutative L^p-spaces. Specific problems include the following: Area I: Let J be an ideal (closed, 2-sided) in a C*-algebra A with A/J separable. Is J Banach complemented in A? An important special case is where J = K, the ideal of compact operators on separable infinite-dimensional Hilbert space; this is related to the uniform approximation property in classical Banach space theory. Area II: Does every separable operator space X with the CSCP completely embed in K? X is said to have the CSCP if it is locally reflexive and completely complemented in every separable locally reflexive superspace. Area III: Is every complemented subspace of a separable nuclear C*-algebra completely isomorphic to a nuclear operator space? Is the converse true? Area IV: Let N be a von Neumann algebra and X be a subspace (infinite- dimensional, linear, and closed) of the predual of N. Does l^p embed in X for some 1 < or = p < or = 2? If M is another von Neumann algebra, and the preduals of M and N are Banach isomorphic, do M and N have the same type? The study of operator spaces involves mathematics underlying the foundation of quantum mechanics. Important "quantized" Banach spaces include C*-algebras such as the Fermion algebra and the algebra of compact operators on Hilbert space, and the preduals of von Neumann algebras, i.e., non-commutative L^1-spaces. The problems concerning the structure of these spaces will be approached from the perspective of classical Banach space theory. This approach has already had considerable success, yielding basic properties of the space of compact operators, and the Banach distinction between the preduals of finite and infinite von Neumann algebras.