Haagerup tensor products and Schur multipliers

In this paper we compare various versions of Schur multipliers: classsical matrix Schur multipliers, discrete Schur multipliers, Schur multipliers with respect to measures and Schur multpliers with respect to spectral measures. The main result of the paper is that in the case of Schur multipliers with respect to measures and with respect to spectral measures the class of such Schur multipliers coincides isometrically with the Haagerup tensor product of the corresponding $L^\infty$ spaces. We deduce this result from a similar familiar fact in the case of discrete Schur multipliers.