A measure-theoretic Grothendieck inequality

Abstract

In this note we develop a notion of integration with respect to a bimeasure μ that allows integration of functions in the projective tensor product L 2 (ν 1 )⊗ L 2 (ν 2 ), where ν 1 and ν 2 are Grothendieck measures for μ. This integral, which agrees with the standard notion of integration with respect to a bimeasure, allows us to integrate inner products and provides a generalization of the Grothendieck inequality to a measure-theoretic setting.

Vector integration and the Grothendieck inequality

“…In an earlier work [3], the author derived a measure-theoretic analog of this theorem: Theorem 1.2. Let (X, A) and (Y, B) be two measurable spaces and H a separable Hilbert space with inner product •, • .…”