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- Operator_algebra abstract "In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings. Although it is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics and quantum field theory.Such algebras can be used to study arbitrary sets of operators with little algebraic relation 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.An operator algebra is typically required to be closed in a specified operator topology inside the algebra of the whole continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties. In some disciplines such properties are axiomized and algebras with certain topological structure become the subject of the research.Though algebras of operators are studied in various contexts (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 separable Hilbert space, endowed with the operator norm topology.In the case of operators on a Hilbert space, the Hermitian adjoint map on operators gives a natural 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*-algebras and von Neumann algebras. 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 valued continuous functions on a locally compact space, or that of measurable functions on a 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.Examples of operator algebras which are not self-adjoint include:nest algebras many commutative subspace lattice algebrasmany limit algebras".
- Operator_algebra wikiPageID "455987".
- Operator_algebra wikiPageRevisionID "558870552".
- Operator_algebra hasPhotoCollection Operator_algebra.
- Operator_algebra subject Category:Functional_analysis.
- Operator_algebra subject Category:Operator_algebras.
- Operator_algebra subject Category:Operator_theory.
- Operator_algebra type Abstraction100002137.
- Operator_algebra type Algebra106012726.
- Operator_algebra type Cognition100023271.
- Operator_algebra type Content105809192.
- Operator_algebra type Discipline105996646.
- Operator_algebra type KnowledgeDomain105999266.
- Operator_algebra type Mathematics106000644.
- Operator_algebra type OperatorAlgebras.
- Operator_algebra type PsychologicalFeature100023100.
- Operator_algebra type PureMathematics106003682.
- Operator_algebra type Science105999797.
- Operator_algebra comment "In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings. Although it is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics and quantum field theory.Such algebras can be used to study arbitrary sets of operators with little algebraic relation simultaneously.".
- Operator_algebra label "Algèbre d'opérateurs".
- Operator_algebra label "Operator algebra".
- Operator_algebra label "Operator-algebra".
- Operator_algebra label "Operatoralgebra".
- Operator_algebra label "Операторная алгебра".
- Operator_algebra label "作用素環論".
- Operator_algebra sameAs Operatoralgebra.
- Operator_algebra sameAs Algèbre_d'opérateurs.
- Operator_algebra sameAs 作用素環論.
- Operator_algebra sameAs Operator-algebra.
- Operator_algebra sameAs m.02bnp_.
- Operator_algebra sameAs Q1892554.
- Operator_algebra sameAs Q1892554.
- Operator_algebra sameAs Operator_algebra.
- Operator_algebra wasDerivedFrom Operator_algebra?oldid=558870552.
- Operator_algebra isPrimaryTopicOf Operator_algebra.