完全正映射的算子值极大f-散度

中文摘要 AI 辅助

我们引入了一种与(0,+∞)上的算子凸函数f相关的、作用于冯·诺依曼代数之间完全正（CP）映射的算子值极大f-散度，该构造的值域为上域冯·诺依曼代数的扩展下半有界自伴部分。我们证明了其计算所用公共CP上界的独立性、联合次可加性、在幺正预复合与正规后复合下的单调性、正规CP映射的鞅收敛定理，以及在点σ-弱拓扑下的联合下半连续性。对于正规正泛函，我们的构造恢复了Hiai的极大f-散度；由此，我们确立了其在任意冯·诺依曼代数上的联合弱下半连续性，解答了Hiai遗留的一个开放问题。当η(t)=t log t时，我们得到算子值Belavkin–Staszewski（BS）相对熵。此外，对于具有σ-有限上域的正规信道，我们证明Hollands与Ranallo的BS信道散度与我们的算子值散度的扩展范数一致。最后，我们给出有限维例子，并得到有限指标条件期望的显式公式；在BS情形下，该公式简化为Jones–Kosaki指标的对数。

英文摘要

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.

发表机构

• Osaka Kyoiku University（大阪教育大学）

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