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# memo_has_moved_text();The Grothendieck inequality revisited

Ron Blei

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 232, Number 1093
ISBNs: 978-0-8218-9855-0 (print); 978-1-4704-1896-0 (online)
DOI: http://dx.doi.org/10.1090/memo/1093
Published electronically: March 11, 2014
Keywords:The Grothendieck inequality, Parseval-like formulas, integral representations, fractional Cartesian products, projective continuity

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Chapters

• Chapter 1. Introduction
• Chapter 2. Integral representations: the case of discrete domains
• Chapter 3. Integral representations: the case of topological domains
• Chapter 4. Tools
• Chapter 5. Proof of Theorem 3.5
• Chapter 6. Variations on a theme
• Chapter 7. More about $\Phi$
• Chapter 8. Integrability
• Chapter 9. A Parseval-like formula for $\langle {\bf {x}}, {\bf {y}}\rangle$, ${\bf {x}} \in l^p$, ${\bf {y}} \in l^q$
• Chapter 10. Grothendieck-like theorems in dimensions $>2$?
• Chapter 11. Fractional Cartesian products and multilinear functionals on a Hilbert space
• Chapter 12. Proof of Theorem 11.11
• Chapter 13. Some loose ends

### Abstract

The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved, concerning continuous functions of two variables over general topological domains. The main result is the construction of a continuous map $\Phi$ from $l^2(A)$ into $L^2(\Omega _A, \mathbb {P}_A)$, where $A$ is a set, $\Omega _A = \{-1,1\}^A$, and $\mathbb {P}_A$ is the uniform probability measure on $\Omega _A$, such that

and

for an absolute constant $K > 1$. ($\Phi$ is non-linear, and does not commute with complex conjugation.) The Parseval-like formula in [[eqref]]ab1 is obtained by iterating the usual Parseval formula in a framework of harmonic analysis on dyadic groups. A modified construction implies a similar integral representation of the dual action between $l^p$ and $l^q$, $\frac {1}{p} + \frac {1}{q} = 1$. Variants of the Grothendieck inequality in higher dimensions are derived. These variants involve representations of functions of $n$ variables in terms of functions of $k$ variables, $0 < k < n.$ Multilinear Parseval-like formulas are obtained, extending the bilinear formula in [[eqref]]ab1. The resulting formulas imply multilinear extensions of the Grothendieck inequality, and are used to characterize the feasibility of integral representations of multilinear functionals on a Hilbert space.