The Grothendieck Inequality Revisited

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 a construction of a continuous map $Φ$ from $l^2(A)$ into $L^2(Ω_A, P_A)$, where $A$ is a set, $Ω_A = {-1,1}^A$, and $P_A$ is the uniform probability measure on $Ω_A$, such that $$\sum_{α\in A} x(α) \bar{y}(α)} = \int_{Ω_A} Φ(x)Φ(\bar{y})dP_A, x \in l^2(A), y \in l^2(A),$$ and $$ |Φ(x)|_{L^{\infty}} \leq K |x|_2, x \in l^2(A),$$ for an absolute constant $K > 1$. ($Φ$ is non-linear, and does not commute with complex conjugation.) The bilinear Parseval-like formula above 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 $l^q$, \ $1/p + 1/q= 1$. Parseval-like formulas are derived in higher dimensions. These variants involve representations of functions of $n$ variables in terms of functions of $k$ variables, $0 < k < n$. Multilinear extensions of the Grothendieck inequality are obtained, and are used to characterize the feasibility of integral representations of multilinear functionals on a Hilbert space.