DECOMPOSABILITY OF BIMODULE MAPS

Consider a unital C*-algebra A, a von Neumann algebra M, a unital sub-C*-algebra C subset of A and a unital *-homomorphism pi: C -> 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 is an element of pi (C)' such that u is valued in eMe and e pi (.)e has a completely positive extension A -> eMe. We also show that this condition is always fulfilled when C has the weak expectation property.