Operator algebra: Difference between revisions

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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 disciplinedisciplines 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.