Completely order bounded maps on non-commutative $${\varvec{L_p}}$$-spaces

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