In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. In particular, it is a set of operators with both algebraic and topological closure properties. Though operator algebras are studied in this generality in the research literature (for example, algebras of pseudo-differential operators acting on spaces of distributions), the term operator algebra is usually used in reference to algebras of bounded operators on a Banach space or, even more specially in reference to algebras of operators on a Hilbert space. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

Such algebras can be used to study arbitrary sets of operators simultaneously. From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general operator algebras are non-commutative rings. In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

In the case of operators on a Hilbert space, the adjoint map on operators gives a natural involution which provides additional algebraic structure which can be imposed on the algebra. In the context of operator algebras on a Hilbert space, the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include C*-algebras and von Neumann algebras. C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication. The Gelfand–Naimark theorem states that an abstract C*-algebra is always isometrically *-isomorphic to a C*-algebra of operators on a Hilbert space. It is possible to give an abstract characterization of a von Neumann algebra as a C*-algebra with a predual. The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, the term adjoint applies in several situations. ... In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ... In mathematics, an element x of a star-algebra is self-adjoint if the involution acts trivially upon it. ... C*-algebras are an important area of research in functional analysis. ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ... In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. ... In mathematics, the predual of an object D is an object P whose dual space is D. For example, the predual of the set of bounded operators is the set of trace class operators. ...

Examples of operator algebras which are not self-adjoint include:

• nest algebras
• many commutative subspace lattice algebras
• many limit algebras

In functional analysis, nest algebras are a class of operator algebras which generalise the upper-triangular matrix algebras to a Hilbert space context. ...

See also

• Topologies on the set of operators on a Hilbert space