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Abstract: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 \(\Phi\sigma_\Omega\Psi\) for completely positive maps dominated by a common ambient map \(\Omega\). The special choice \(\Omega=\Phi+\Psi\) 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.
From: Mohsen Kian [view email]
[v1]
Tue, 12 May 2026 07:58:03 UTC (17 KB)
[v2]
Mon, 18 May 2026 13:14:38 UTC (18 KB)
[v3]
Tue, 2 Jun 2026 11:40:01 UTC (1 KB) (withdrawn)
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