Proof

Proof. Assume (i), fix ${\tau }_A\in D({\mathcal{H}}_A)$, and define $\theta (X)=\Theta ({\tau }_A\otimes X)$. This is a channel. Property (i) proves (ii) on states, hence on all operators by linear spanning.

The adjoint of ${\mathrm{Tr}}_A$ is $Y\mapsto I_A\otimes Y$, so (ii) implies (iii). Conversely, (iii) says that every output expectation is determined by the $C$-reduction, proving (i).

For an untouched reference $E$, the factorization extends as

If ${\mathcal{J}}_A$ is a channel, then ${\mathrm{Tr}}_A[({\mathcal{J}}_A\otimes {\mathrm{id}}_{CE})({\rho }_{ACE})]={\mathrm{Tr}}_A{\rho }_{ACE}$, as follows by testing against observables on $C\otimes E$ and using ${\mathcal{J}}_A^{\ast }(I_A)=I_A$.