Multidimensional operator multipliers

Abstract

We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C ∗ C^* -algebras satisfying certain boundedness conditions. In the case of commutative C ∗ C^* -algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to some given representations of the corresponding C ∗ C^* -algebras do not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding C ∗ C^* -algebras.