Factorization of linear operators and geometry of Banach spaces
Author(s)
Bibliographic Information
Factorization of linear operators and geometry of Banach spaces
（Regional conference series in mathematics, no. 60）
Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, c1986
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"Lectures presented at the NSF-CBMS regional conference, University of Missouri-Columbia, June 25-29, 1984"--T.p. verso
Includes bibliographies
Description and Table of Contents
Description
This book surveys the considerable progress made in Banach space theory as a result of Grothendieck's fundamental paper ""Resume De la Theorie Metrique des Produits Tensoriels Topologiques"". The author examines the central question of which Banach spaces $X$ and $Y$ have the property that every bounded operator from $X$ to $Y$ factors through a Hilbert space, in particular when the operators are defined on a Banach lattice, a $C^*$-algebra or the disc algebra and $H^\infty$. He reviews the six problems posed at the end of Grothendieck's paper, which have now all been solved (except perhaps the exact value of Grothendieck's constant), and includes the various results which led to their solution. The last chapter contains the author's construction of several Banach spaces such that the injective and projective tensor products coincide; this gives a negative solution to Grothendieck's sixth problem.Although the book is aimed at mathematicians working in functional analysis, harmonic analysis and operator algebras, its detailed and self-contained treatment makes the material accessible to nonspecialists with a grounding in basic functional analysis. In fact, the author is particularly concerned to develop very recent results in the geometry of Banach spaces in a form that emphasizes how they may be applied in other fields, such as harmonic analysis and $C^*$-algebras.
Table of Contents
Absolutely summing operators and basic applications Factorization through a Hilbert space Type and cotype. Kwapien's theorem The ""abstract"" version of Grothendieck's theorem Grothendieck's theorem Banach spaces satisfying Grothendieck's theorem Applications of the volume ratio method Banach lattices $C^*$-algebras Counterexamples to Grothendieck's conjecture.
by "Nielsen BookData"