Decomposability of Bimodule Maps - Merdy, Christian Le; Oliveira, Lina (MATHEMATICA SCANDINAVICA, 2016)
Consider a unital $C^*$-algebra $A$, a von Neumann algebra $M$, a unital sub-$C^*$-algebra $C\subset A$ and a unital $*$-homomorphism $π\colon C→ M$. Let $u\colon A→ M$ 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 $e\in π(C)'$ such that $u$ is valued in $\mathit{e\mkern0.5muMe}$ and $eπ({·})e$ has a completely positive extension $A→ \mathit{e\mkern0.5muMe}$. We also show that this condition is always fulfilled when $C$ has the weak expectation property.