Operator space

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title: "Operator space" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["banach-spaces", "operator-theory"] topic_path: "general/banach-spaces" source: "https://en.wikipedia.org/wiki/Operator_space" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In functional analysis, a discipline within mathematics, an operator space is a normed vector space (not necessarily a Banach space) "given together with an isometric embedding into the space B(H) of all bounded operators on a Hilbert space H.". The appropriate morphisms between operator spaces are completely bounded maps.

Equivalent formulations

Equivalently, an operator space is a subspace of a C*-algebra.

Category of operator spaces

The category of operator spaces includes operator systems and operator algebras. For operator systems, in addition to an induced matrix norm of an operator space, one also has an induced matrix order. For operator algebras, there is still the additional ring structure.

References

References

1. Paulsen, Vern. (2002). "Completely Bounded Maps and Operator Algebras". [[Cambridge University Press]].
2. Pisier, Gilles. (2003). "Introduction to Operator Space Theory". [[Cambridge University Press]].
3. (2004). "Operator Algebras and Their Modules: An Operator Space Approach". [[Oxford University Press]].

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