Contributes to the axiomatization of set theory, developing a system of axioms that avoids the paradoxes of naive set theory. His approach is more elegant than Zermelo-Fraenkel set theory in some respects, though less widely adopted.

Makes significant contributions to operator theory, particularly the study of von Neumann algebras (also known as W"-algebras), which are fundamental objects in functional analysis and have applications in quantum mechanics. Von Neumann algebras provide the basis for non-commutative geometry.

Proves the mean ergodic theorem, a fundamental result in ergodic theory that has applications in dynamical systems and statistical mechanics. This theorem provides information about the long-term average behavior of dynamical systems.

Develops the theory of continuous geometry, a generalization of projective geometry that allows for dimensions to take continuous values. This work is a significant contribution to abstract algebra and geometry.

Known for his exceptional mathematical rigor and ability to solve complex problems using sophisticated mathematical techniques. He was a prolific mathematician who contributed to a wide range of mathematical fields.