Decomposability of Bimodule Maps

Consider a unital $C^*$-algebra $A$, a von Neumann algebra $M$, a unital sub-$C^*$-algebra $C\subset A$ and a unital $*$-homomorphism $\pi\colon C\to M$. Let $u\colon A\to M$ be a decomposable map (i.e. a linear combination of completely positive maps) which is a $C$-bimodule map with respect to $\pi$. We show that $u$ is a linear combination of $C$-bimodule completely positive maps if and only if there exists a projection $e\in \pi(C)'$ such that $u$ is valued in $\mathit{e\mkern0.5muMe}$ and $e\pi({\cdot})e$ has a completely positive extension $A\to \mathit{e\mkern0.5muMe}$. We also show that this condition is always fulfilled when $C$ has the weak expectation property.