Operator-valued maximal f-divergences for completely positive maps
Abstract
We introduce an operator-valued maximal f-divergence for completely positive (CP) maps between von Neumann algebras, associated with an operator convex function f on (0,+infinity). The construction takes values in the extended lower-semibounded self-adjoint part of the codomain von Neumann algebra. We prove independence of the common CP upper bound used in its computation, joint subadditivity, monotonicity under unital precomposition and normal postcomposition, a martingale convergence theorem for normal CP maps, and joint lower semicontinuity in the point-sigma-weak topology. For normal positive functionals, our construction recovers Hiai's maximal f-divergence. As a consequence, we establish its joint weak lower semicontinuity for arbitrary von Neumann algebras, answering a question left open by Hiai. For eta(t)=t log t, we obtain an operator-valued Belavkin--Staszewski (BS) relative entropy. Moreover, for normal channels with sigma-finite codomain, we prove that the BS channel divergence of Hollands and Ranallo coincides with the extended norm of our operator-valued divergence. Finally, we give finite-dimensional examples and obtain an explicit formula for a finite-index conditional expectation. In the BS case, this formula reduces to the logarithm of the Jones--Kosaki index.
Related
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- Banach-Cesaro positive operator-valued measure
- A Separable Hilbertian Operator Space with CBAP and No Completely Bounded Basis
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- Sufficient positive maps between von Neumann algebras: Rényi divergences
BibTeX
@misc{okayasu2026operatorvalued,
title = {{Operator-valued maximal $f$-divergences for completely positive maps}},
author = {Rui Okayasu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00554},
eprint = {2609.00554},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00554},
}
RIS
TY - UNPB
TI - Operator-valued maximal f-divergences for completely positive maps
AU - Rui Okayasu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00554
UR - https://arxiv.org/abs/2609.00554
AB - We introduce an operator-valued maximal f-divergence for completely positive (CP) maps between von Neumann algebras, associated with an operator convex function f on (0,+infinity). The construction takes values in the extended lower-semibounded self-adjoint part of the codomain von Neumann algebra. We prove independence of the common CP upper bound used in its computation, joint subadditivity, monotonicity under unital precomposition and normal postcomposition, a martingale convergence theorem for normal CP maps, and joint lower semicontinuity in the point-sigma-weak topology. For normal positive functionals, our construction recovers Hiai's maximal f-divergence. As a consequence, we establish its joint weak lower semicontinuity for arbitrary von Neumann algebras, answering a question left open by Hiai. For eta(t)=t log t, we obtain an operator-valued Belavkin--Staszewski (BS) relative entropy. Moreover, for normal channels with sigma-finite codomain, we prove that the BS channel divergence of Hollands and Ranallo coincides with the extended norm of our operator-valued divergence. Finally, we give finite-dimensional examples and obtain an explicit formula for a finite-index conditional expectation. In the BS case, this formula reduces to the logarithm of the Jones--Kosaki index.
ER -