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Cyclic And Separating Vector
In mathematics, the notion of a cyclic and separating vector is important in the theory of von Neumann algebras, and in particular in Tomita–Takesaki theory. A related notion is that of a vector which is cyclic for a given operator. The existence of cyclic vectors is guaranteed by the Gelfand–Naimark–Segal (GNS) construction. Definitions Given a Hilbert space ''H'' and a linear space ''A'' of bounded linear operators in ''H'', an element Ω of ''H'' is said to be ''cyclic'' for ''A'' if the linear space ''A''Ω = is norm-dense in ''H''. The element Ω is said to be ''separating'' if ''a''Ω = 0 with ''a'' in ''A'' implies ''a'' = 0. * Any element Ω of ''H'' defines a semi-norm ''p'' on A by ''p''(''a'') = , , ''a''Ω, , . Saying that Ω is separating is equivalent with saying that ''p'' is actually a norm. * If Ω is cyclic for ''A'' then it is separating for the commutant ''A′'', which is the von Neumann algebra of all bounded operators in ''H'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomita–Takesaki Theory
In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a good structure theory for these previously intractable objects. The theory was introduced by , but his work was hard to follow and mostly unpublished, and little notice was taken of it until wrote an account of Tomita's theory. Modular automorphisms of a state Suppose that ''M'' is a von Neumann algebra acting on a Hilbert space ''H'', and Ω is a cyclic and separating vector of ''H'' of norm 1. (Cyclic means that ''MΩ'' is dense in ''H'', and separating means that the map from ''M'' to ''MΩ'' is injective.) We write \phi for the vector state \phi(x) = (x\Omega, \Omega) of ''M'', so that ''H'' is constructed from \phi using the Gelfand–Naimark–Segal construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Vector
An operator ''A'' on an (infinite dimensional) Banach space or Hilbert space H has a cyclic vector ''f'' if the vectors ''f'', ''Af'', ''A2f'',... span H. Equivalently, ''f'' is a cyclic vector for ''A'' in case the set of all vectors of the form ''p(A)f'', where ''p'' varies over all polynomials, is dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ... in H. See also * Cyclic and separating vector References Abstract algebra Functional analysis {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand–Naimark–Segal Construction
In functional analysis, a discipline within mathematics, given a C*-algebra ''A'', the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of ''A'' and certain linear functionals on ''A'' (called ''states''). The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal. States and representations A *-representation of a C*-algebra ''A'' on a Hilbert space ''H'' is a mapping π from ''A'' into the algebra of bounded operators on ''H'' such that * π is a ring homomorphism which carries involution on ''A'' into involution on operators * π is nondegenerate, that is the space of vectors π(''x'') ξ is dense as ''x'' ranges through ''A'' and ξ ranges through ''H''. Note that if ''A'' has an identity, nondegeneracy means exactly π is unit-preserving, i.e. π maps the identity of ''A'' to the identity operator on ''H''. A state on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Linear 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 spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \, Lx\, _Y \leq M \, x\, _X. The smallest such M is called the operator norm of L and denoted by \, L\, . A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called " bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-norm
In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms. Definition Let X be a vector space over either the real numbers \R or the complex numbers \Complex. A real-valued function p : X \to \R is called a if it satisfies the following two conditions: # Subadditivity/Triangle inequality: p(x + y) \leq p(x) + p(y) for all x, y \in X. # Absolute homogeneity: p(s x) =, s, p(x) for all x \in X and all scalars s. These two conditions imply that p(0) = 0If z \in X denotes the zero vector in X while 0 denote the zero scalar, then absolute homogeneity implies that p(z) = p(0 z) = , 0, p(z) = 0 p(z) = 0. \blacksquare and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "\,\leq\," in the homogeneity axiom. It can also re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \, Lx\, _Y \leq M \, x\, _X. The smallest such M is called the operator norm of L and denoted by \, L\, . A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called " bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
*-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings and , where is commutative and has the structure of an associative algebra over . Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert's space and Hermitian adjoints. However, it may happen that an algebra admits no involution. Definitions *-ring In mathematics, a *-ring is a ring with a map that is an antiautomorphism and an involution. More precisely, is required to satisfy the following properties: * * * * for all in . This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and fourth axioms, making it redundant. Elements such that are called '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Adjoint
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \langle \cdot,\cdot \rangle is the inner product on the vector space. The adjoint may also be called the Hermitian conjugate or simply the Hermitian after Charles Hermite. It is often denoted by in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators are represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose). The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H. The definition has been further extended to include unbounded '' densely defined'' operators whose domain is topologically dense in—but not necessarily equal to— ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Linear Functional
In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, \leq) is a linear functional f on V so that for all positive elements v \in V, that is v \geq 0, it holds that f(v) \geq 0. In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When V is a complex vector space, it is assumed that for all v\ge0, f(v) is real. As in the case when V is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace W\subseteq V, and the partial order does not extend to all of V, in which case the positive elements of V are the positive elements of W, by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any x \in V equal to s^s for some s \in V to a real nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
