Meaning
Detector readout is the compression of a regional operator to the chosen detector carrier.
For the detector embedding J and a bounded regional operator A, the readout is C(A)=J∗AJ. This records what A looks like through that detector. Multiplication need not survive compression: C(AB)−C(A)C(B)=J∗A(I−JJ∗)BJ measures excursions through states the detector omits.
Example (illustrative)
Take Jz=(z,0) in C² and A=B=[[0,1],[1,0]]. Then C(A)=C(B)=0, but AB=I and C(AB)=1. Acting twice leaves the detector subspace and returns; compressing each action separately loses that route.
Scope and limits
Readout is not automatically an algebra homomorphism. Multiplicativity needs the detector projection to reduce the relevant algebra, or an equivalent verified condition.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 7.2.