Pinching, anchoring, and dilation

For a qubit set $A$, let

be computational-basis pinching on $A$, with identity factors elsewhere. It is CPTP, idempotent, and Hilbert–Schmidt self-adjoint. For $J\subseteq \mathcal{N}$, define the fixed channel

Thus ${\Delta }_J$ is defined on all operators, not just tower states.