# Operator space

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Operator_space
> Markdown URL: https://mediated.wiki/source/Operator_space.md
> Source: https://en.wikipedia.org/wiki/Operator_space
> Source revision: 1153496369
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [functional analysis](/source/Functional_analysis), a discipline within mathematics, an **operator space** is a [normed vector space](/source/Normed_vector_space) (not necessarily a [Banach space](/source/Banach_space))[1] "given together with an [isometric](/source/Isometry) [embedding](/source/Injective_map) into the space *B(H)* of all [bounded operators](/source/Bounded_operator) on a [Hilbert space](/source/Hilbert_space) *H*.".[2][3] The appropriate morphisms between operator spaces are [completely bounded maps](https://en.wikipedia.org/w/index.php?title=Completely_bounded_map&action=edit&redlink=1).

## Equivalent formulations

Equivalently, an operator space is a [subspace](/source/Linear_subspace) of a [C*-algebra](/source/C*-algebra).

## Category of operator spaces

The [category](/source/Category_(mathematics)) of operator spaces includes [operator systems](/source/Operator_system) and [operator algebras](/source/Operator_algebra). 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](/source/Ring_(mathematics)) structure.

## See also

- [Gilles Pisier](/source/Gilles_Pisier)

- [Operator system](/source/Operator_system)

## References

1. **[^](#cite_ref-1)** Paulsen, Vern (2002). [*Completely Bounded Maps and Operator Algebras*](https://books.google.com/books?id=VtSFHDABxMIC). [Cambridge University Press](/source/Cambridge_University_Press). p. 26. [ISBN](/source/ISBN_(identifier)) [978-0-521-81669-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-81669-4). Retrieved 2022-03-08.

1. **[^](#cite_ref-2)** Pisier, Gilles (2003). [*Introduction to Operator Space Theory*](https://books.google.com/books?id=0pKL-o7WUOAC&dq=Operator+space&pg=PA1). [Cambridge University Press](/source/Cambridge_University_Press). p. 1. [ISBN](/source/ISBN_(identifier)) [978-0-521-81165-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-81165-1). Retrieved 2008-12-18.

1. **[^](#cite_ref-3)** Blecher, David P.; Christian Le Merdy (2004). [*Operator Algebras and Their Modules: An Operator Space Approach*](https://books.google.com/books?id=lwprbgvFA4IC&dq=%22Operator+space%22&pg=PP11). [Oxford University Press](/source/Oxford_University_Press). First page of Preface. [ISBN](/source/ISBN_(identifier)) [978-0-19-852659-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-852659-9). Retrieved 2008-12-18.

v t e Functional analysis (topics – glossary) Spaces Banach Besov Fréchet Hilbert Inner product Normed Nuclear Schwartz Sobolev Topological vector Properties Barrelled Quasi-barrelled Complete Dual (Algebraic / Topological) Locally convex Reflexive Separable Theorems Hahn–Banach Riesz representation Closed graph Uniform boundedness principle Kakutani fixed-point Krein–Milman Min–max Gelfand–Naimark Banach–Alaoglu Operators Adjoint Bounded Compact Hilbert–Schmidt Normal Nuclear Trace class Transpose Unbounded Unitary Algebras Banach algebra C*-algebra Spectrum of a C*-algebra Operator algebra Group algebra of a locally compact group Von Neumann algebra Open problems Invariant subspace problem Mahler's conjecture Applications Hardy space Spectral theory of ordinary differential equations Heat kernel Index theorem Calculus of variations Functional calculus Integral linear operator Jones polynomial Topological quantum field theory Noncommutative geometry Riemann hypothesis Distribution (or Generalized functions) Advanced topics Approximation property Balanced set Choquet theory Weak topology Banach–Mazur distance Tomita–Takesaki theory Category

v t e Spectral theory and *-algebras Basic concepts Involution/*-algebra Banach algebra B*-algebra C*-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C*-algebra Spectral radius Operator space Main results Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras Special Elements/Operators Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit Spectrum Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap Decomposition Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa Spectral Theorem Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras Special algebras Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory Finite-Dimensional Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem Generalizations Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary) Miscellaneous Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator Examples Wiener algebra Applications Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

---
Adapted from the Wikipedia article [Operator space](https://en.wikipedia.org/wiki/Operator_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Operator_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.