  We continue the study of multidimensional operator multipliers initiated in
[arXiv:math/0701645]. 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.
