2011/08/15 by Gilles Pisier · 11 citations
Mathematics · Computer Science · #Advanced Operator Algebra Research #Complexity and Algorithms in Graphs #Limits and Structures in Graph Theory #Mathematics #Banach space #Pure mathematics #Algebra over a field #Tensor product
paper · pdf · doi:10.1090/s0273-0979-2011-01348-9
published in Bulletin of the American Mathematical Society 49(2), 237-323 (American Mathematical Society)
openalex publication_date 2011/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Probably the most famous of Grothendieckâs contributions to Banach space theory is the result that he himself described as âthe fundamental theorem in the metric theory of tensor productsâ. That is now commonly referred to as âGrothendieckâs theoremâ (âGTâ for short), or sometimes as âGrothendieckâs inequalityâ. 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 âoperator spacesâ or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bellâs 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 âsemidefinite programmingâ 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 authorâs 1986 CBMS notes.