Relative Kubo-Ando Means of Completely Positive Maps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913139753222144 |
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| author | Kian, Mohsen |
| author_facet | Kian, Mohsen |
| contents | We develop a Kubo--Ando theory on order intervals of completely positive maps. Using Arveson's Radon--Nikodym theorem as a structural tool, we define relative Kubo--Ando means \(Φσ_ΩΨ\) for completely positive maps dominated by a common ambient map \(Ω\). The special choice \(Ω=Φ+Ψ\) yields an intrinsic mean of two completely positive maps.
We prove that these means are independent of the chosen Stinespring representation and satisfy the expected order-theoretic properties, including monotonicity, transformer inequalities, Jensen-type inequalities, data processing, and monotonicity with respect to the ambient map. For the geometric mean, we obtain a block-positivity characterization and show that the intrinsic geometric mean vanishes exactly when the two maps have no nonzero common completely positive submap. Finally, we compare the construction with existing finite-dimensional and form-theoretic approaches: for maps between matrix algebras it agrees with the Choi-matrix mean, and in the geometric case it agrees with Okayasu's Pusz--Woronowicz mean on their common domain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_11701 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relative Kubo-Ando Means of Completely Positive Maps Kian, Mohsen Operator Algebras Functional Analysis 47A64, 46L07, 47A63, 47B65 We develop a Kubo--Ando theory on order intervals of completely positive maps. Using Arveson's Radon--Nikodym theorem as a structural tool, we define relative Kubo--Ando means \(Φσ_ΩΨ\) for completely positive maps dominated by a common ambient map \(Ω\). The special choice \(Ω=Φ+Ψ\) yields an intrinsic mean of two completely positive maps. We prove that these means are independent of the chosen Stinespring representation and satisfy the expected order-theoretic properties, including monotonicity, transformer inequalities, Jensen-type inequalities, data processing, and monotonicity with respect to the ambient map. For the geometric mean, we obtain a block-positivity characterization and show that the intrinsic geometric mean vanishes exactly when the two maps have no nonzero common completely positive submap. Finally, we compare the construction with existing finite-dimensional and form-theoretic approaches: for maps between matrix algebras it agrees with the Choi-matrix mean, and in the geometric case it agrees with Okayasu's Pusz--Woronowicz mean on their common domain. |
| title | Relative Kubo-Ando Means of Completely Positive Maps |
| topic | Operator Algebras Functional Analysis 47A64, 46L07, 47A63, 47B65 |
| url | https://arxiv.org/abs/2605.11701 |