Operator space

In functional analysis, a discipline within mathematics, an operator space is a normed vector space (not necessarily a Banach space)^[1] "given together with an isometric embedding into the space B(H) of all bounded operators on a Hilbert space H.".^[2]^[3] 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

^ Paulsen, Vern (2002). Completely Bounded Maps and Operator Algebras. Cambridge University Press. p. 26. ISBN 978-0-521-81669-4. Retrieved 2022-03-08.
^ Pisier, Gilles (2003). Introduction to Operator Space Theory. Cambridge University Press. p. 1. ISBN 978-0-521-81165-1. Retrieved 2008-12-18.
^ Blecher, David P.; Christian Le Merdy (2004). Operator Algebras and Their Modules: An Operator Space Approach. Oxford University Press. First page of Preface. ISBN 978-0-19-852659-9. Retrieved 2008-12-18.

[1] Paulsen, Vern (2002). Completely Bounded Maps and Operator Algebras. Cambridge University Press. p. 26. ISBN 978-0-521-81669-4. Retrieved 2022-03-08.

[2] Pisier, Gilles (2003). Introduction to Operator Space Theory. Cambridge University Press. p. 1. ISBN 978-0-521-81165-1. Retrieved 2008-12-18.

[3] Blecher, David P.; Christian Le Merdy (2004). Operator Algebras and Their Modules: An Operator Space Approach. Oxford University Press. First page of Preface. ISBN 978-0-19-852659-9. Retrieved 2008-12-18.

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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

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