Von Neumann algebras are a special type of mathematical structure used in functional analysis and quantum mechanics. They are defined as a set of bounded operators on a Hilbert space that are closed under taking adjoints and contain the identity operator. This makes them useful for studying the properties of quantum systems. These algebras can be classified into different types, such as type I, type II, and type III, based on their structural properties. They play a crucial role in the mathematical foundation of quantum theory, providing a framework for understanding observables and states in quantum mechanics.