In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, a discipline within mathematics, an operator space is a
normed vector space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914ā18), but struggled in the inter-war ...
(not necessarily a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
) "given together with an
isometric
The term ''isometric'' comes from the Greek for "having equal measurement".
isometric may mean:
* Cubic crystal system, also called isometric crystal system
* Isometre, a rhythmic technique in music.
* "Isometric (Intro)", a song by Madeon from ...
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ...
into the space ''B(H)'' of all
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
s on a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
''H''.".
The appropriate morphisms between operator spaces are
completely bounded maps.
Equivalent formulations
Equivalently, an operator space is a
subspace of a
C*-algebra
In mathematics, specifically in functional analysis, a Cā-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...
.
Category of operator spaces
The
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
of operator spaces includes
operator system
Given a unital C*-algebra \mathcal , a *-closed subspace ''S'' containing ''1'' is called an operator system. One can associate to each subspace \mathcal \subseteq \mathcal of a unital C*-algebra an operator system via S:= \mathcal+\mathcal ...
s and
operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study o ...
s. 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
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
structure.
See also
*
Gilles Pisier
Gilles I. Pisier (born 18 November 1950) is a professor of mathematics at the Pierre and Marie Curie University and a distinguished professor and A.G. and M.E. Owen Chair of Mathematics at the Texas A&M University. He is known for his contribution ...
*
Operator system
Given a unital C*-algebra \mathcal , a *-closed subspace ''S'' containing ''1'' is called an operator system. One can associate to each subspace \mathcal \subseteq \mathcal of a unital C*-algebra an operator system via S:= \mathcal+\mathcal ...
References
{{SpectralTheory
Banach spaces
Operator theory