Operator algebra: Difference between revisions

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{{Short description|Branch of functional analysis}}

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]], [[differential geometry]], [[quantum statistical mechanics]] and [[quantum field theory]].

{{Use American English|date = March 2019}}

{{Ring theory sidebar}}

In [[functional analysis]], a branch of [[mathematics]], 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 [[function composition|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 [[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.

The results obtained in the study of operator algebras are often phrased in [[algebra]]ic terms, while the techniques used are often highly [[mathematical analysis|analytic]].<ref>''Theory of Operator Algebras I'' By [[Masamichi Takesaki]], Springer 2012, p vi</ref> Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to [[representation theory]], [[differential geometry]], [[quantum statistical mechanics]], [[quantum information]], and [[quantum field theory]].

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.

==Overview==

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.

SuchOperator 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 [[noncommutative ring|non-[[commutative]] [[ringRing (mathematics)|ringrings]]s.

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.

An operator algebra is typically required to be [[closure (mathematics)|closed]] in a specified operator [[topology]] inside the whole algebra of the whole continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties. In some disciplinedisciplines such properties are [[axiom|axiomized]]atized and algebras with certain topological structure become the subject of the research.

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.

Though algebras of operators are studied in various contextcontexts (for example, algebras of [[pseudo-differential operator]]s acting on spaces of [[Distribution (mathematics)|distributions]]), the term ''operator algebra'' is usually used in reference to algebras of [[bounded operator]]s on a [[Banach space]] or, even more speciallyspecifically in reference to algebras of operators on a [[Separable space|separable]] [[Hilbert space]], endowed with the operator [[normoperator (mathematics)|norm]] topology.

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

*[[nest algebra]]s

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 whichthat 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, and [[AW*-algebra]]s. C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication. Such abstractly defined C*-algebraalgebras can be identified to a certain closed [[Subalgebra#Subalgebras for algebras over a ring or field|subalgebra]] of the algebra of the continuous linear operators on a suitable Hilbert space. A similar result holds for von Neumann algebras.

*many [[commutative subspace lattice algebra]]s

*many [[limit algebra]]s

[[commutative algebra|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 functionsfunction]]s 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.

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

*[[nest algebra]]s ,

*many [[commutative subspace lattice algebra]]s,

*many [[limit algebra]]s .

==See also==

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

* {{annotated link|Banach algebra}}

== References ==

* {{annotated link|Matrix mechanics}}

*{{cite book

[[* {{annotated link|Topologies on the set of operators on a Hilbert space]]}}

* {{annotated link|Vertex operator algebra}}

== References ==

{{reflist}}

==Further reading==

* {{cite book

| last = Blackadar

| first = Bruce

Line 27 ⟶ 44:

| series = Encyclopaedia of Mathematical Sciences

| year = 2005

| isbn = 35402848693-540-28486-9 }}

* M. Takesaki, ''Theory of Operator Algebras I'', Springer, 2001.

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