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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Grothendieck’s Theorem, past and present

Author: Gilles Pisier
Journal: Bull. Amer. Math. Soc. 49 (2012), 237-323
MSC (2010): Primary 46B28, 46L07; Secondary 46B85, 81P40
Published electronically: August 12, 2011
MathSciNet review: 2888168
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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” for 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. This expository paper (where many proofs are included), presents 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.

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Additional Information

Gilles Pisier
Affiliation: Texas A&M University, College Station, Texas 77843
MR Author ID: 140010

Received by editor(s): January 26, 2011
Received by editor(s) in revised form: March 31, 2011
Published electronically: August 12, 2011
Additional Notes: Partially supported by NSF grant 0503688
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.