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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    $$2p&lt; q &lt; \infty $$  2 p &lt; q &lt; ∞ 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.

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