Infinite-Dimensional Banach and Operator Spaces

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ABSTRACTOdell This project involves several open problems in geometric functional analysis. 1) Does there exist a distortable Banach space of bounded distortion? 2) If all spreading models of any block basis of a given basis for X are 1-equivalent to some l_p space does X contain l_p? 3) Must every Banach space admit a "nice" spreading model? 4) The study of the ordinal l_1(+) index of a Banach space. In the usual geometry of space one can measure the distance between two points by means of a carpenter's tape measure (distance is measured "as the crow flies"). There are however other natural distances which a mathematician encounters. For example the "taxicab distance" between two points on this surface would be the distance between the points as measured by a taxi that could only travel horizontally or vertically (imagine all roads run N-S or E-W). Mathematicians must study the geometry of these different notions of distance in order to solve those problems from whence the geometry arose. Furthermore many mathematical problems require one to work in spaces of higher than three dimensions, even infinite dimensional space (for example in quantum mechanics differential equations,...). This project concerns such problems. For example the author and Th. Schlumprecht proved that ordinary (tape measure) infinite dimensional space could have its geometry altered ("distorted") so as to be that one could not recapture the original geometry up to any given multiple in any infinite dimensional "slice". This is false for the taxicab geometry. But it remains a famous open problem as to whether or not there exists a geometry which can be altered but not too badly. Other problems of this project involve determining infinite dimensional substructures of a geometry given limited "asymptotical" (less than infinite dimensional but more than finite dimensional) information.