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

==Overview==

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 non-[[commutative]] [[ring (mathematics)|ring]]s.

An operator algebra is typically required to be [[closure (mathematics)|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 [[axiom]]ized and algebras with certain topological structure become the subject of the research.