Grothendieck's Theorem, past and present

TL;DR

This paper reviews the development and applications of Grothendieck's theorem across various fields, highlighting recent advances in operator spaces and its impact on quantum mechanics, graph theory, and computer science.

Contribution

It provides a comprehensive overview of Grothendieck's theorem's evolution, emphasizing recent developments and interdisciplinary applications beyond classical Banach space theory.

Findings

Recent formulation of GT in operator spaces

GT's role in quantum mechanics and Bell's inequality

Application of GT in polynomial-time algorithms for NP-hard problems

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.