NeumannAlgebren

Neumann algebren, also spelled von Neumann algebras, are a class of operator algebras in functional analysis named after John von Neumann. A von Neumann algebra M is a unital -subalgebra of the bounded operators B(H) on a Hilbert space H that is closed in the weak operator topology (WOT), equivalently in the strong operator topology. By the bicommutant theorem, M equals its double commutant M'' (the commutant M' consists of all operators that commute with every element of M). Von Neumann algebras are also dual spaces, possessing a unique predual M_ such that M is the dual of M_.

Commutative von Neumann algebras are isomorphic to L∞-spaces of measure spaces, with the algebra acting by multiplication on L2(X, μ). This establishes a functional-analytic framework for measure theory: abelian von Neumann algebras correspond to measure-theoretic objects.

A central part of the theory is the classification of von Neumann algebras into types I, II, and III, and the notion of factors (algebras with trivial center). Type I algebras are built from B(H) for some Hilbert space H. Type II factors admit a trace, with Type II1 having a finite, faithful, normal trace, and Type II∞ being semifinite. Type III factors do not admit any nonzero finite trace and arise prominently in modular theory.

Key examples include B(H) itself, abelian examples L∞(X, μ) acting on L2(X, μ), and group von Neumann algebras L(G) generated by the left regular representation of a group G. Von Neumann algebras sit at the intersection of functional analysis, representation theory, and mathematical physics, and are closely related to C-algebras via representations and the construction of enveloping von Neumann algebras A''. They play a foundational role in quantum mechanics, statistical mechanics, and noncommutative geometry.