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Finite-Corner Reconstruction and Separation for Cones of Normal Maps

  • Mohsen Kian
  • Fuad Kittaneh
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Abstract

We develop a finite-corner framework for convex cones of normal maps between ℬ(ℋ) and ℬ(𝒦). The basic structural assumption is stability under finite cut--pad operations. We prove that every point-ultraweakly closed cut--pad stable cone is completely determined by its finite-dimensional corner cones, and conversely that every coherent family of closed finite-dimensional corner cones admits a unique global realization of this type. For a cut--pad stable cone that is not necessarily point-ultraweakly closed, reconstruction from the norm closures of its corner cones yields exactly its point-ultraweak closure. Combining this reconstruction principle with finite-dimensional separation and the Choi representation, we show that non-membership in a global cone is always detected on a single finite-dimensional corner by a finite-dimensional witness. We illustrate the framework for completely positive and decomposable maps; in the decomposable case, the Hermitian parts of the finite-corner witnesses are PPT.

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Cite

BibTeXRIS
BibTeX
@misc{kian2026finitecorner,
  title = {{Finite-Corner Reconstruction and Separation for Cones of Normal Maps}},
  author = {Mohsen Kian and Fuad Kittaneh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04889},
  eprint = {2609.04889},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04889},
}
RIS
TY  - UNPB
TI  - Finite-Corner Reconstruction and Separation for Cones of Normal Maps
AU  - Mohsen Kian
AU  - Fuad Kittaneh
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04889
UR  - https://arxiv.org/abs/2609.04889
AB  - We develop a finite-corner framework for convex cones of normal maps between ℬ(ℋ) and ℬ(𝒦). The basic structural assumption is stability under finite cut--pad operations. We prove that every point-ultraweakly closed cut--pad stable cone is completely determined by its finite-dimensional corner cones, and conversely that every coherent family of closed finite-dimensional corner cones admits a unique global realization of this type. For a cut--pad stable cone that is not necessarily point-ultraweakly closed, reconstruction from the norm closures of its corner cones yields exactly its point-ultraweak closure. Combining this reconstruction principle with finite-dimensional separation and the Choi representation, we show that non-membership in a global cone is always detected on a single finite-dimensional corner by a finite-dimensional witness. We illustrate the framework for completely positive and decomposable maps; in the decomposable case, the Hermitian parts of the finite-corner witnesses are PPT.
ER  -