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Abelian Von Neumann Algebra
In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The prototypical example of an abelian von Neumann algebra is the algebra ''L''∞(''X'', μ) for μ a σ-finite measure on ''X'' realized as an algebra of operators on the Hilbert space ''L''2(''X'', μ) as follows: Each ''f'' ∈ ''L''∞(''X'', μ) is identified with the multiplication operator : \psi \mapsto f \psi. Of particular importance are the abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants. Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure spaces (''X'', μ) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maharam's Classification Theorem
In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval ,1on the reals), and "purely atomic parts", using the counting measure on some discrete space. The theorem is due to Dorothy Maharam. It was extended to localizable measure spaces by Irving Segal. The result is important to classical Banach space theory, in that, when considering the Banach space given as an Lp space of measurable functions over a general measurable space, it is sufficient to understand it in terms of its decomposition into non-atomic and atomic parts. Maharam's theorem can also be translated into the language of abelian von Neumann algebras. Every abelian von Neumann algebra is isomorphic to a product of σ-finite abelian von Neumann algebras, and every σ-finite ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon–Nikodym Theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space. One way to derive a new measure from one already given is to assign a density to each point of the space, then integrate over the measurable subset of interest. This can be expressed as :\nu(A) = \int_A f \, d\mu, where is the new measure being defined for any measurable subset and the function is the density at a given point. The integral is with respect to an existing measure , which may often be the canonical Lebesgue measure on the real line or the ''n''-dimensional Euclidean space (corresponding to our sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicontinuous
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Integral
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series ''On Rings of Operators''. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings. Results on direct integrals can be viewed as generalizations of results about finite-dimensional C*-algebras of matrices; in this case the results are easy to prove directly. The infinite-dimensional case is complicated by measure-theoretic technicalities. Direct integral theory was als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Integral
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series ''On Rings of Operators''. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings. Results on direct integrals can be viewed as generalizations of results about finite-dimensional C*-algebras of matrices; in this case the results are easy to prove directly. The infinite-dimensional case is complicated by measure-theoretic technicalities. Direct integral theory was als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicity Theory
In abstract algebra, multiplicity theory concerns the multiplicity of a module ''M'' at an ideal ''I'' (often a maximal ideal) :\mathbf_I(M). The notion of the multiplicity of a module is a generalization of the degree of a projective variety. By Serre's intersection formula, it is linked to an intersection multiplicity in the intersection theory. The main focus of the theory is to detect and measure a singular point of an algebraic variety (cf. resolution of singularities). Because of this aspect, valuation theory, Rees algebras and integral closure are intimately connected to multiplicity theory. Multiplicity of a module Let ''R'' be a positively graded ring such that ''R'' is finitely generated as an ''R''0-algebra and ''R''0 is Artinian. Note that ''R'' has finite Krull dimension ''d''. Let ''M'' be a finitely generated ''R''-module and ''F''''M''(''t'') its Hilbert–Poincaré series. This series is a rational function of the form :\frac, where P(t) is a polynomial. By ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-atomic Measure
In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless. Definition Given a measurable space (X, \Sigma) and a measure \mu on that space, a set A\subset X in \Sigma is called an atom if \mu(A) > 0 and for any measurable subset B \subset A with \mu(B) of A are atoms, and /math> is called an atomic class. If \mu is a \sigma-finite measure, there are countably many atomic classes. Examples * Consider the set ''X'' = and let the sigma-algebra \Sigma be the power set of ''X''. Define the measure \mu of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons , for ''i'' = 1, 2, ..., 9, 10 is an atom. * Consider the Lebesgue measure on the real line. This measure has no atoms. Atomic measures A \sigma-finite measure \mu on a measurable space (X, \Sigma) is called atomic o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Folklore
In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists. More specifically, folk mathematics, or mathematical folklore, is the body of theorems, definitions, proofs, facts or techniques that circulate among mathematicians by word of mouth, but have not yet appeared in print, either in books or in scholarly journals. Quite important at times for researchers are folk theorems, which are results known, at least to experts in a field, and are considered to have established status, though not published in complete form. Sometimes, these are only alluded to in the public literature. An example is a book of exercises, described on the back cover: Another distinct category is well-knowable mathematics, a term introduced by John Conway. These mathematical matters are known and factual, but not in active circulation in relation with curren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Operator Topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is continuous for any vectors x and y in the Hilbert space. Explicitly, for an operator T there is base of neighborhoods of the following type: choose a finite number of vectors x_i, continuous functionals y_i, and positive real constants \varepsilon_i indexed by the same finite set I. An operator S lies in the neighborhood if and only if , y_i(T(x_i) - S(x_i)), 0. Relationships between different topologies on ''B(X,Y)'' The different terminology for the various topologies on B(X,Y) can sometimes be confusing. For instance, "strong convergence" for vectors in a normed space sometimes refers to norm-convergence, which is very often distinct from (and stronger than) than SOT-convergence when the normed space in question is B(X,Y). The w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
