Grothendieck's Theorem, Past and Present
Grothendieck’s Theorem, past and present by Gilles Pisier∗ Texas A&M University College Station, TX 77843, U. S. A. and Universit´eParis VI Equipe d’Analyse, Case 186, 75252 Paris Cedex 05, France January 26, 2011 Abstract Probably the most famous of Grothendieck’s contributions to Banach space theory is the result that he himself described as “the fundamental theorem in the metric theory of tensor products”. That is now commonly referred to as “Grothendieck’s theorem” (GT in short), or sometimes as “Grothendieck’s inequality”. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C∗-algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of “operator spaces” or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell’s inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by “semidefinite programming’ and hence solved in polynomial time. In this expository paper, we present a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author’s 1986 CBMS notes. ∗Partially supported by NSF grant 0503688 Contents 1 Introduction 1 2 Classical GT 7 3 The Grothendieck constants 15 4 The Hahn-Banach argument 17 5 The “little” GT 20 6 Banach spaces satisfying GT and operator ideals 22 7 Non-commutative GT 23 8 Non-commutative “little GT” 25 9 Non-commutative Khintchine inequality 26 10 Maurey factorization 32 11 Best constants (Non-commutative case) 33 12 C∗-algebra tensor products, Nuclearity 35 13 Operator spaces, c.b.
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