We introduce a noncommutative version of Schur multipliers relative to an operator
ideal. In this setting the functions of two variables are replaced by elements from a
tensor product of C*-algebras, and the measures (or spectral measures) by
representations. For commutative C*-algebras this approach agrees with Birman and
Solomyak’s theory of double operator integrals. We study the dependence of
the spaces of multipliers on the choice of representations and find that the
question is closely related to Voiculescu and Arveson’s theory of approximately
equivalent representations. The space of multipliers universal with respect
to the chosen measures is related to the Haagerup tensor product of the
algebras.
