The predual of the space of decomposable maps from a -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 ∈ M k ( A ) , y ∈ M l ( R ∗ ) and (2) v ⊗ w is positive if v ∈ M k ( A ) + and w ∈ M l ( R ∗ ) + .