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Though algebras of operators are studied in various contexts (for example, algebras of [[pseudo-differential operator]]s acting on spaces of distributions), the term ''operator algebra'' is usually used in reference to algebras of [[bounded operator]]s on a Banach space or, even more specially in reference to algebras of operators on a separable [[Hilbert space]], endowed with the operator [[norm (mathematics)|norm]] topology.
In the case of operators on a Hilbert space, the [[Hermitian adjoint]] map on operators gives a natural [[Involution (mathematics)|involution]] which provides an additional algebraic structure which can be imposed on the algebra. In this context, the best studied examples are [[self-adjoint]] operator algebras, meaning that they are closed under taking adjoints. These include [[C*-algebra]]s and [[von Neumann algebra]]s. C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication. Such abstractly defined C*-algebras can be identified to a certain closed subalgebra of the algebra of the continuous linear operators on a suitable Hilbert space. A similar result holds for von Neumann algebras.
Commutative self-adjoint operator algebras can be regarded as the algebra of [[Complex numbers|complex]] valued continuous functions on a [[locally compact space]], or that of measurable functions on a [[measurable space|standard measurable space]]. Thus, general operator algebras are often regarded as a noncommutative generalizations of these algebras, or the structure of the ''base space'' on which the functions are defined. This point of view is elaborated as the philosophy of [[noncommutative geometry]], which tries to study various non-classical and/or pathological objects by noncommutative operator algebras.
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