AI 中文总结
该研究针对半有限von Neumann代数，建立带等号条件的Jensen迹不等式，解答相关问题，推导非交换Lamperti型不等式，刻画一类F-范数非交换Orlicz空间上的线性等距，得到对应非交换Lamperti定理。
AI 中文摘要
设$\boldsymbol{\tau}$为半有限忠实正规迹的半有限von Neumann代数$\boldsymbol{\tau}$。我们建立了全一般性的Jensen迹不等式并刻画其等号情形，解答了[Kosaki2013]与[HaradaKosaki2008]提出的两个问题。作为应用，我们推导了非交换Lamperti型不等式及其等号情形，利用该结果刻画了一类F-范数非交换Orlicz空间上的线性等距（不一定是满射），得到了线性等距的非交换Lamperti定理。
英文摘要
Let $\mathcal M$ be a semifinite von Neumann algebra equipped with a semifinite faithful normal trace $τ$. We establish Jensen's trace inequality in full generality and characterize its equality case, which answers two questions raised in [Kosaki2013] and [HaradaKosaki2008]. As an application, we derive noncommutative Lamperti-type inequalities and their equality conditions. Employing this result, we characterize linear isometries (not necessarily surjective) on a class of $F$-normed noncommutative Orlicz spaces, which provides a noncommutative Lamperti's theorem for linear isometries.
