An operator algebra is an algebra of linear operators on a vector space, most often a norm-closed or topology-closed subalgebra of the bounded operators on a Hilbert space. Multiplication is operator composition and is generally noncommutative.
The two principal classes are C*-algebras, which are closed in the operator norm and under the adjoint, 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.
