The predual of the space of decomposable maps from a $C^{\ast }$-algebra into a von Neumann algebra

For a $C^{\ast }$-algebra $\mathcal{A}$ and a von Neumann algebra $\mathcal{R}$, we describe the predual of space $D(\mathcal{A},\mathcal{R})$ of decomposable maps from $\mathcal{A}$ into $\mathcal{R}$ equipped with decomposable norm. This predual is found to be the matrix regular operator space structure on $\mathcal A \otimes \mathcal R_$ 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 $\mathcal A \otimes \mathcal R_$underthetwonaturalrestrictions:(1)$|x \otimes y| \le |x| |y|$for$x\in M_k(\mathcal A), y \in M_l(\mathcal R_)$and(2)$v \otimes w$ispositiveif$v \in M_k(\mathcal A)^+$and$w \in M_l(\mathcal R_)^+$.