Relative Kubo-Ando Means of Completely Positive Maps

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 \(Φσ_ΩΨ\) 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.

This paper has been withdrawn by the author due to discovering overlap with the recent literature. The manuscript is currently being fundamentally restructured to focus on the ambient independence of relative operator connections