Abstract
In this paper, we study determinantal inequalities in von Neumann algebras. Such inequalities arise naturally in several areas of mathematics and have applications, for example, in image processing. We first extend determinantal inequalities for positive matrices established by Ghabries, Abbas, and Mourad (2020) to the setting of finite von Neumann algebras by using the Fuglede--Kadison determinant. We then derive several related inequalities based on logarithmic submajorization and determinant functionals associated with generalized singular numbers. Furthermore, we generalize results of Yan and Han (2022) from finite to semifinite von Neumann algebras, obtaining new inequalities for positive $\tau$-measurable operators. These results broaden the scope of previously known determinantal inequalities and contribute to the theory of operator inequalities in noncommutative analysis.