Jensen's trace inequality with equality condition and noncommutative Lamperti's theorem

Abstract

Let $\mathcal M$ be a semifinite von Neumann algebra equipped with a semifinite faithful normal trace $\tau$. 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.