Abstract:We introduce the geometric mean and the parallel sum of completely positive (CP) maps between 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.
From: Rui Okayasu [view email]
[v1]
Thu, 7 May 2026 11:14:32 UTC (31 KB)
[v2]
Tue, 1 Sep 2026 01:31:49 UTC (30 KB)
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