Operator algebra
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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.
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 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, but this is somewhat more technical.
Examples of operator algebras which are not self-adjoint include: