Operator algebra: Difference between revisions

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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]].

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]].

==Overview==

Operator 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 [[noncommutative ring|non-commutative]] [[Ring (mathematics)|rings]].

An operator algebra is typically required to be [[closure (mathematics)|closed]] in a specified operator [[topology]] inside the whole algebra of continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties. In some disciplines such properties are [[axiom]]izedatized 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 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 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 that 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, [[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*-algebras 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.

[[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 function]]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.