Mathematical Sciences: Structure of Infinite-Dimensional Banach Spaces

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9500874 Rosenthal The proposers intend to work on several fundamental problems in the structure theory of infinite dimensional Banach spaces. Specifically, Odell will study whether there are Banach spaces of bounded distortion, or of arbitrary distortion but not biorthogonal distortion. These questions are strongly motivated by Odell's recent joint discovery with Schlumprecht, that Hilbert space is distortable in the strongest sense. He also intends to study whether certain almost isometric asymptotic assumptions on a Banach space insure that it contains one of the classical spaces lp for 1 or = p infinity or c subscript 0. Odell will also investigate whether the Huff criterion characterizes spaces isomorphic to those with the uniform Kadec-Klee property, and whether the Point of Continuity Property is basically determined. Rosenthal will study further possible characterizations of spaces containing c subscript 0, related to his recent characterization that a non-reflexive space contains c subscript 0 provided all its subspaces have weakly sequentially complete duals. For example, he will study the question of whether an script L infinity saturated space contains c subscript 0, and whether a Banach subspace X of C(K) contains c subscript 0 provided the complement of X in X** contains functions whose alpha-th oscillation is bounded, for all countable ordinals alpha. Rosenthal also wises to investigate Banach spaces with separable duals, for example the problem of whether such spaces are isomorphic to subspaces of quotients of C(alpha) (for alpha a countable ordinal), with particular attention given to the Hilbert space case. Rosenthal will also study the long-standing open problem of whether complemented subspaces of C(K) spaces are again isomorphic to a C(K)-space. The study of structural properties of infinite-dimensional Banach spaces is one of the most profound, fundamental research areas i n modern analysis. These spaces underlie the mathematical foundations of quantum mechanics, and understanding and penetrating their structure is basic for modern research in functional analysis and its applications to physics. Recently there have been spectacular advances in this area. Deep long standing open problems have been solved, and both proposers have been heavily involved in these advances. Of course there are many more fundamental open problems remaining and the proposers intend to tackle some of these basic questions. ***