Operator algebra: Difference between revisions

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In [[functional analysis]], an '''operator algebra''' is an [[algebra over a field|algebra]] of [[continuous function (topology)|continuous]] [[linear operator]]s 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]] and, [[differential geometry]], [[quantum statistical mechanics]] and [[quantum field theory]].

Such algebras can be used to study [[wiktionary:arbitrary|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]] [[ring (mathematics)|ring]]s.

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 discipline such properties are [[axiom|axiomized]] and algebras with certain topological structure become the subject of the research.

Though algebras of operators are studied in various context (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 [[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*-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*-algebra 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 measuremeasurable space]]. Thus, general operator algebras are often regarded as a non commutativenoncommutative 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:

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==See also==

*[[Topologies on the set of operators on a Hilbert space]]

== References ==

*[[Topologies on the set of operators on a Hilbert space]]

*{{cite book

| last = Blackadar

| first = Bruce

| title = Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

| publisher = [[Springer-Verlag]]

| series = Encyclopaedia of Mathematical Sciences

| year = 2005

| isbn = 3540284869 }}

[[Category:Operator theory]]

[[Category:Functional analysis]]

[[fr:Algèbre d'opérateurs]]