Relative Kubo-Ando Means of Completely Positive Maps

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 $\mathrm{\Phi }{\sigma }_{\mathrm{\Omega }}\mathrm{\Psi }$ for completely positive maps dominated by a common ambient map $\mathrm{\Omega }$. The special choice $\mathrm{\Omega }=\mathrm{\Phi }+\mathrm{\Psi }$ 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.