Isometries between C$^*$-algebras with finite corank

What the math gives to ML

The paper gives an explicit family of operator-norm isometric linear maps between matrix algebras, built from Jordan-type representations involving both X and X^t, contractive compression matrices, unitary changes of basis, and an arbitrary contractive finite-dimensional remainder. The transferable asset is exact preservation of the spectral norm without restricting the layer to a conventional orthogonal map. This suggests matrix-valued neural layers with guaranteed non-expansiveness or exact operator-norm preservation, potentially improving stability in deep residual, recurrent, and state-space architectures. The first implementation should use the defect-rank construction as a constrained matrix layer and compare it against ordinary spectral normalization and orthogonal initialization.