Grothendieck’s Theorem, past and present
Academic Article

 Overview

 Research

 Identity

 Additional Document Info

 View All

Overview
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 noncommutative 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. © 2011 American Mathematical Society.
altmetric score
author list (cited authors)
citation count
publication date
publisher
published in
Research
keywords

46b28, 46b07

Mathph

Math.fa

Math.mp

Math.oa
Identity
Digital Object Identifier (DOI)
Additional Document Info
start page
end page
volume
issue