Decomposability of bimodule maps

Abstract

Consider a unital C*-algebra A, a von Neumann algebra M, a unital sub-C*-algebra C of A and a unital *-homomorphism $\pi$ 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 $\pi$. We show that u is a linear combination of C-bimodule completely positive maps if and only if there exists a projection e in the commutant of $\pi(C)$ such that u is valued in eMe and $e\pi(.)e$ has a completely positive extension A --> eMe.