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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^* +\mathbb 1 . The appropriate morphisms between operator systems are completely positive maps. By a theorem of Choi and Effros, operator systems can be characterized as *-vector spaces equipped with an Archimedean matrix order.Choi M.D., Effros, E.G. Injectivity and operator spaces. Journal of Functional Analysis 1977 See also * Operator space 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 ' ... References Operator theory Operator algebras {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unital Algebra
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity inst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C*-algebra
In mathematics, specifically in functional analysis, a C∗-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 continuous linear operators on a complex Hilbert space with two additional properties: * ''A'' is a topologically closed set in the norm topology of operators. * ''A'' is closed under the operation of taking adjoints of operators. Another important class of non-Hilbert C*-algebras includes the algebra C_0(X) of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, linear subspaces, flats, and affine subspaces are also called ''linear manifolds'' for emphasizing that there are also manifolds. is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a ''subspace'' when the context serves to distinguish it from other types of subspaces. Definition If ''V'' is a vector space over a field ''K'' and if ''W'' is a subset of ''V'', then ''W'' is a linear subspace of ''V'' if under the operations of ''V'', ''W'' is a vector space over ''K''. Equivalently, a nonempty subset ''W'' is a subspace of ''V'' if, whenever are elements of ''W'' and are elements of ''K'', it follows that is in ''W''. As a corollary, all vector spaces are equipped with at least two ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Positive Map
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition. Definition Let A and B be C*-algebras. A linear map \phi: A\to B is called positive map if \phi maps positive elements to positive elements: a\geq 0 \implies \phi(a)\geq 0. Any linear map \phi:A\to B induces another map :\textrm \otimes \phi : \mathbb^ \otimes A \to \mathbb^ \otimes B in a natural way. If \mathbb^\otimes A is identified with the C*-algebra A^ of k\times k-matrices with entries in A, then \textrm\otimes\phi acts as : \begin a_ & \cdots & a_ \\ \vdots & \ddots & \vdots \\ a_ & \cdots & a_ \end \mapsto \begin \phi(a_) & \cdots & \phi(a_) \\ \vdots & \ddots & \vdots \\ \phi(a_) & \cdots & \phi(a_) \end. We say that \phi is k-positive if \textrm_ \otimes \phi is a positive map, and \phi is called completely positive if \phi is k-positive for all k. Properties * Positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Space
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. 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 contributi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
