Proposition: Algebra target and Heisenberg inclusion

The map ${\mathcal{R}}_{\mathcal{M}}$ is CPTP. Its adjoint, restricted to block-diagonal output operators, is

Consequently $\mathrm{ran}{\mathcal{R}}_{\mathcal{M}}^{†}=\mathcal{M}$.

Proof. Fix orthonormal bases $\{|k{⟩}_{\alpha }\}$ of $S_{\alpha }$. Kraus operators are

where ${\iota }_{\alpha }:C_{\alpha }\to K$ is the output inclusion. Their adjoint products sum to $I_H$. The adjoint formula follows by taking traces. Off-diagonal output blocks have zero adjoint image.