Refined von Neumann-type trace inequality and its application

The purpose of this note is to establish logarithmic submajorization inequalities that are associated with von Neumann's trace inequality for operators in finite von Neumann algebras. As an application, we extend some of of the results in Carlen and Lieb [Some matrix rearrangement inequalities. Ann Mat Pura Appl. 2006;185:S315–S324. doi: 10.1007/s10231-004-0147-z] and Chayes [Matrix rearrangement inequalities revisited; 2021. arXiv:2009.04032; Reverse Hölder, Minkowski, and Hanner inequalities for matrices; 2021. arXiv:2103.09915] to non-commutative Lp spaces associated with finite von Neumann algebras. Finally, we provide an estimation of the geodesic distance on P.