Operator algebra: Difference between revisions

Revision as of 13:58, 27 September 2024

Latest revision as of 10:35, 19 July 2025

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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]]atized 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.