The predual of the space of decomposable maps from a C∗-algebra into a von Neumann algebra

For a C∗-algebra A and a von Neumann algebra R, we describe the predual of space D(A,R) of decomposable maps from A into R equipped with decomposable norm. This predual is found to be the matrix regular operator space structure on A⊗R∗ with a certain universal property. Its matrix norms are the largest and its positive cones on each matrix level are the smallest among all possible matrix regular operator space structures on A⊗R∗ under the two natural restrictions: (1) ‖x⊗y‖≤‖x‖‖y‖ for x∈Mk(A),y∈Ml(R∗) and (2) v⊗w is positive if v∈Mk(A)+ and w∈Ml(R∗)+.