Completely Order Bounded Maps on Non-Commutative $L_p$-Spaces

Abstract: We define norms on $L_p(\mathcal{M}) \otimes M_n$ where $\mathcal{M}$ is a von Neumann algebra and $M_n$ is the complex $n \times n$ matrices. We show that a linear map $T: L_p(\mathcal{M}) \to L_q(\mathcal{N})$ is decomposable if $\mathcal{N}$ is an injective von Neumann algebra, the maps $T \otimes Id_{M_n}$ have a common upper bound with respect to our defined norms, and $p = \infty$ or $q = 1$. For $2p < q < \infty$ we give an example of a map $T$ with uniformly bounded maps $T \otimes Id_{M_n}$ which is not decomposable.