Completely positive completion of partial matrices whose entries are completely bounded maps

We study completion problems of partial matrices associated with a graph where entries are completely bounded maps on aC ^*-algebra. We characterize a graph for which every -partial completely positive matrix has a completely positive completion. As a special case we study -partial functional matrices. We give a necessary and sufficient condition for a -partial functional matrix to have a positive completion and a representation for such matrices. These generalize some results on inflated Schur product maps due to Paulsen, Power and Smith. As an application, we study completely positive completions of partial matrices whose entries are completely bounded multipliers of the Fourier algebra of a locally compact group.