A measure-theoretic Grothendieck inequality (Q984719)

The following generalized Grothendieck inequality (for the classical one, take \(X=Y=\mathbb{N}\)) is the main result of the paper: Let \((X,\mathcal{A})\) and \((Y,\mathcal{B})\) be measurable spaces and \(H\) a separable Hilbert space over the reals with inner product \(\langle\cdot,\cdot\rangle\). If \(f\) and \(g\) are bounded measurable \(H\)-valued functions, the function \(\langle f,g\rangle:X\times Y\to\mathbb{R}\) defined by \[ \langle f,g\rangle (x,y)=\langle f(x),g(x)\rangle \] can be integrated with respect to any bimeasure \(\mu\) on \(\mathcal{A}\times\mathcal{B}\) and \[ \left|\int\langle f,g\rangle\;d\mu\right|\leq K_G\|f\|_\infty\|g\|_\infty\|\mu\|, \] where \(K_G\) is the Grothendieck constant. The integral is given by \(\int\langle f,g\rangle\;d\mu=\lim_{N\to\infty}\int\left(\sum_{j=1}^N f_j\otimes g_j\right)\;d\mu\), where \((f_j)\) and \((g_j)\) are the coordinate functions with respect to an orthonormal basis of \(H\). It is shown that this integral is independent of the choice of basis and that it coincides with the standard integral when \(\langle f,g\rangle\in L^{\infty}(X)\hat\otimes L^{\infty}(Y)\).