Determine the nesting of adjoint coarse-graining ranges for generic coarse-grained models

Determine whether the ranges of the adjoint coarse-graining maps satisfy the nesting property \({\rm Ran}[\mathcal I_k^*] \subset {\rm Ran}[\mathcal I_{k+1}^*]\) for every consecutive pair of scales in a generic coarse-grained model, equivalently whether \({\rm Ker}[\mathcal I_{k+1}] \subset {\rm Ker}[\mathcal I_k]\).

Background

The proposed refinement maps require isometric embeddings between coarse Hilbert spaces that also satisfy transitivity. A direct construction using $\mathcal I_m\mathcal I_k^*$ fails because the adjoint coarse-graining maps are not generally partial isometries and because the required nesting of their ranges has not been established.

The authors introduce iterated coarse-graining maps $\mathcal C_k$ to enforce a related kernel-nesting property and then use polar decompositions to construct refinement maps. Thus, the unresolved question concerns the original coarse-graining maps $\mathcal I_k$ for a generic model, rather than the validity of the alternative construction actually adopted.