Gilles Pisier, "Factorization of Linear Operators and Geometry of Banach Spaces "
English | ISBN: 0821807102 | | 154 pages | PDF | 5 MB
This book surveys the considerable progress made in Banach space theory as a result of Grothendieck's fundamental paper Resumé de la théorie métrique 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.
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