Operator algebra: Difference between revisions

Revision as of 21:37, 25 September 2024

Revision as of 13:58, 27 September 2024

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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 specially in reference to algebras of operators on a [[Separable space|separable]] [[Hilbert space]], endowed with the [[operator norm]] topology.