The predual of the space of decomposable maps from a $C^*$-algebra into a von Neumann algebra

Abstract

For a $C^*$-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 $\mathcal A \otimes \mathcal R_*$ under the two natural restrictions: (1) $|x \otimes y| \le |x| |y|$ for $x\in M_k(\mathcal A), y \in M_l(\mathcal R_*)$ and (2) $v \otimes w$ is positive if $v \in M_k(\mathcal A)^+$ and $w \in M_l(\mathcal R_*)^+$.