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 Hölder type generalized singular numbers inequality.

中文翻译：

---

有限冯诺依曼代数中算子的对数次大化不等式

本文的目的是研究有限冯诺依曼代数中算子的对数次大化不等式。首先，一些由 Garg 和 Aulja 引起的对数次大化不等式被扩展到有限冯诺依曼代数中的算子的情况。作为应用，我们得到了在这种情况下算子的一些新的 Fuglede-Kadison 行列式不等式。其次，我们改进并推广到有限冯诺依曼代数的设置，一个广义 Hölder 型广义奇异数不等式。