In mathematics, von Neumann algebras are self-adjoint operator algebras that are closed under a chosen operator topology. When the underlying Hilbert space is finite-dimensional, the von Neumann algebra is said to be a finite-dimensional von Neumann algebra. The finite-dimensional case differs from the general von Neumann algebras in that topology plays no role and they can be characterized using Wedderburn's theory of semisimple algebras.