An operator algebra is an algebra of linear operators on a vector space, most often a subalgebra of the bounded operators on a Hilbert space that is closed in the norm topology or another topology. Multiplication is operator composition and generally need not be commutative.
The two principal classes are C*-algebras, which are closed in the operator norm and under the adjoint operator, and von Neumann algebras, which are closed in a topology defined by their action on the Hilbert space and contain the identity operator. Operator algebras connect functional analysis, operator theory, and noncommutative geometry.
