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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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Grothendieck’s Theorem, past and present
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by Gilles Pisier PDF
Bull. Amer. Math. Soc. 49 (2012), 237-323 Request permission


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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 49 (2012), 237-323
  • MSC (2010): Primary 46B28, 46L07; Secondary 46B85, 81P40
  • DOI:
  • MathSciNet review: 2888168