Logarithmic submajorisations inequalities for operators in a finite von Neumann algebra

TL;DR

This paper extends logarithmic submajorisation inequalities and Fuglede-Kadison determinant inequalities to operators in finite von Neumann algebras, generalizing existing results and improving inequalities in this setting.

Contribution

It introduces new logarithmic submajorisation inequalities and generalizes a H"{o}lder type inequality for finite von Neumann algebra operators.

Findings

Extended Garg and Aulja inequalities to finite von Neumann algebras

Derived new Fuglede-Kadison determinant inequalities

Generalized H"{o}lder type inequalities for singular numbers

Abstract

The aim of this paper is to study the logarithmic submajorisations inequalities for operators in a finite von Neumann algebra. Firstly, some logarithmic submajorisations inequalities due to Garg and Aulja are extended to the case of operators in a finite von Neumann algebra. As an application, we get some new Fuglede-Kadison determinant inequalities of operators in that circumstance. Secondly, we improve and generalize to the setting of finite von Neumann algebras, a generalized H\"{o}lder type generalized singular numbers inequality.