This is an announcement for the paper "The Grothendieck inequality revisited" by Ron Blei.
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 representation of the inner product in a Hilbert space by an integral with uniformly bounded and continuous integrands. The Parseval-like formula 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 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 Parseval-like formulas are obtained, extending the bilinear formula. The resulting formulas yield multilinear extensions of the bilinear Grothendieck inequality, and are used to characterize the feasibility of integral representations of multilinear functionals on a Hilbert space, within a class of functionals whose kernels are supported by fractional Cartesian products.
Archive classification: math.FA
Submitted from: blei@math.uconn.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1111.7304
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