Grothendieck's Theorem, past and present

By Gilles Pisier. 2011

237 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/42d08e6983e5e10c1b871386f72e259111c790d4

How researchers see this paper

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In Computational Theory and Mathematics

1. Javier Parcet
2. Marius Junge
3. Tao Mei
4. Nathanial P. Brown Narutaka Ozawa
5. Gilles Pisier
6. M. Kennedy
7. Xin Li
8. Yusuke Isono

In Discrete Mathematics and Combinatorics

1. Tao Mei
2. Marius Junge
3. Javier Parcet
4. Nathanial P. Brown Narutaka Ozawa
5. Adrian Ioana
6. J. Peterson

In Mathematical Physics

1. Javier Parcet
2. Tao Mei
3. Marius Junge
4. Nathanial P. Brown Narutaka Ozawa
5. Quanhua Xu
6. Gilles Pisier
7. Narcisse Randrianantoanina
8. Uffe Haagerup

Citations in the corpus

• About the Connes Embedding Conjecture---Algebraic approaches--- (2012, background): “active. It is known that this map is bounded (by KC G<1.41). In case this 22 NARUTAKAOZAWA map is moreover completely contractive, the theorem easily follows from Pisier’s trick. See Section 12 in [Pi2] for more information. Proof. We first prove the ‘if’ part. For notational simplicity, denote C=”
• Smooth Fourier multipliers on group von Neumann algebras (2010, methodology): “By the little Grothendieck inequality [55, 61], this means that v is absolutely 2-summing so that we can find a factorization v = w ◦ jξ where jξ is the map f ∈ L∞(Ω1, μ1) 7→ fξ ∈ L2(Ω1, μ1) for some ξ with ∫ Ω1 |ξ| dμ1 = 1 and we have ‖w‖ ≤ 2 √ π ‖v‖.”