The Isomorphic Structure of Banach and Operator Spaces

点击翻译按钮获取中文摘要

ABSTRACT: This proposal deals with problems concerning the isomorphic structureof 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/Jseparable. Is J Banach complemented in A? An important special case iswhere J = K, the ideal of compact operators on separable infinite-dimensionalHilbert space; this is related to the uniform approximation property inclassical Banach space theory.Area II: Does every separable operator space X with the CSCP completelyembed in K? X is said to have the CSCP if it is locally reflexive andcompletely complemented in every separable locally reflexive superspace.Area III: Is every complemented subspace of a separable nuclear C*-algebracompletely 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 inX for some 1 or = p or = 2? If M is another von Neumann algebra, and thepreduals of M and N are Banach isomorphic, do M and N have the same type? The study of operator spaces involves mathematics underlying thefoundation of quantum mechanics. Important "quantized" Banach spaces includeC*-algebras such as the Fermion algebra and the algebra of compact operatorson Hilbert space, and the preduals of von Neumann algebras, i.e.,non-commutative L^1-spaces. The problems concerning the structure of thesespaces will be approached from the perspective of classical Banach spacetheory. This approach has already had considerable success, yielding basicproperties of the space of compact operators, and the Banach distinctionbetween the preduals of finite and infinite von Neumann algebras.