Decomposability of bimodule maps

Consider a unital C*-algebra A, a von Neumann algebra M, a unital sub-C*-algebra C of A and a unital *-homomorphism $π$ from C to M. Let u: A --> M be a decomposable map (i.e. a linear combination of completely positive maps) which is a C-bimodule map with respect to $π$. We show that u is a linear combination of C-bimodule completely positive maps if and only if there exists a projection e in the commutant of $π(C)$ such that u is valued in eMe and $eπ(.)e$ has a completely positive extension A --> eMe.

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