Geometric Means and Lebesgue-type Decomposition of Completely Positive Maps

We introduce the geometric mean and the parallel sum of completely positive (CP) maps on von Neumann algebras, based on the Pusz–Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM–GM–HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi–Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo–Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando’s decomposition of bounded positive operators and Kosaki’s decomposition of normal positive functionals on von Neumann algebras.