Compactness properties of operator multipliers

We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C^*-algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C^*-algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C^*-Algebra, Diplomarbeit, Universität des Saarlandes, Saarbrücken, 1982].