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Noncommutative Torus
In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori ''A''θ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra of continuous functions on the 2-torus. Many topological and geometric properties of the classical 2-torus have algebraic analogues for the noncommutative tori, and as such they are fundamental examples of a noncommutative space in the sense of Alain Connes. Definition For any real number ''θ'', the noncommutative torus A_\theta is the C*-subalgebra of B(L^2(\mathbb/\mathbb)), the algebra of bounded linear operators of square-integrable functions on the unit circle S^1 \subset \mathbb, generated by two unitary operators U, V\begin U(f)(z) &= e^f(z) \\ V(f)(z) &= f(z - \theta) \endwhere e^ is the parametrization of the point z on the circle \mathbb/\mathbb in \mathbb. A quick calculation shows that ''VU'' = ''e''−2π ''i'' θ''UV''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Rotation
In the mathematical theory of dynamical systems, an irrational rotation is a map : T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1 where ''θ'' is an irrational number. Under the identification of a circle with R/Z, or with the interval , 1with the boundary points glued together, this map becomes a rotation of a circle by a proportion ''θ'' of a full revolution (i.e., an angle of 2''πθ'' radians). Since ''θ'' is irrational, the rotation has infinite order in the circle group and the map ''T''''θ'' has no periodic orbits. Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map : T_\theta :S^1 \to S^1, \quad \quad \quad T_\theta(x)=xe^ The relationship between the additive and multiplicative notations is the group isomorphism : \varphi:( ,1+) \to (S^1, \cdot) \quad \varphi(x)=xe^. It can be shown that is an isometry. There is a strong distinction in circle rotations th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Linear Transformations
In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transformation is a ''transformation'' that is represented by a ''fraction'' whose numerator and denominator are ''linear''. In the most basic setting, , and are complex numbers (in which case the transformation is also called a Möbius transformation), or more generally elements of a field. The invertibility condition is then . Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. When are integer (or, more generally, belong to an integral domain), is supposed to be a rational number (or to belong to the field of fractions of the integral domain. In this case, the invertibility condition is that must be a unit of the domain (that is or in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morita Equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules are additively equivalent (denoted by _M\approx_M). It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958. Motivation Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings. Every ring ''R'' has a natural ''R''-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules is more general and gives useful information. Because of this, one often studies a ring by studying the category of modules over that ring. Morita equivalence takes this viewpoint to a natural conclusion by defining rings to be Morita equivalent if their module categories are equivalent. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' then ''a'' + ''g'' ≤ ''b'' + ''g'' and ''g'' +'' a'' ≤ ''g'' +'' b''. An element ''x'' of ''G'' is called positive if 0 ≤ ''x''. The set of elements 0 ≤ ''x'' is often denoted with ''G''+, and is called the positive cone of ''G''. By translation invariance, we have ''a'' ≤ ''b'' if and only if 0 ≤ -''a'' + ''b''. So we can reduce the partial order to a monadic property: if and only if For the general group ''G'', the existence of a positive cone specifies an order on ''G''. A group ''G'' is a partially orderable group if and only if there exists a subset ''H'' (which is ''G''+) of ''G'' such that: * 0 ∈ ''H'' * if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a'' + ''b'' ∈ ''H'' * if ''a'' ∈ ''H'' then -''x'' + ''a'' + ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator K-theory
In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras. Overview Operator K-theory resembles topological K-theory more than algebraic K-theory. In particular, a Bott periodicity theorem holds. So there are only two K-groups, namely ''K''0, which is equal to algebraic ''K''0, and ''K''1. As a consequence of the periodicity theorem, it satisfies excision. This means that it associates to an extension of C*-algebras to a long exact sequence, which, by Bott periodicity, reduces to an exact cyclic 6-term-sequence. Operator K-theory is a generalization of topological K-theory, defined by means of vector bundles on locally compact Hausdorff spaces. Here, a vector bundle over a topological space ''X'' is associated to a projection in the C* algebra of matrix-valued—that is, M_n(\mathbb)-valued—continuous functions over ''X''. Also, it is known that isomorphism of vector bundles transla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear C*-algebra
In the mathematical field of functional analysis, a nuclear C*-algebra is a C*-algebra A such that the injective and projective C*- cross norms on A \oplus B are the same for every C*-algebra B. This property was first studied by under the name "Property T", which is not related to Kazhdan's property T. Characterizations Nuclearity admits the following equivalent characterizations: * The identity map, as a completely positive map, approximately factors through matrix algebras. By this equivalence, nuclearity can be considered a noncommutative analogue of the existence of partitions of unity. * The enveloping von Neumann algebra is injective. * It is amenable as a Banach algebra. * It is isomorphic to a C*-subalgebra B of the Cuntz algebra \mathcal_2 with the property that there exists a conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it woul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State (functional Analysis)
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both . Density matrices in turn generalize state vectors, which only represent pure states. For ''M'' an operator system in a C*-algebra ''A'' with identity, the set of all states of'' ''M, sometimes denoted by S(''M''), is convex, weak-* closed in the Banach dual space ''M''*. Thus the set of all states of ''M'' with the weak-* topology forms a compact Hausdorff space, known as the state space of ''M'' . In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number). Jordan decomposition States can be viewed as noncommutative generalizations of probability measures. By Gel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Algebra Of A Locally Compact Group
In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group. The algebra ''Cc''(''G'') of continuous functions with compact support If ''G'' is a locally compact Hausdorff group, ''G'' carries an essentially unique left-invariant countably additive Borel measure ''μ'' called a Haar measure. Using the Haar measure, one can define a convolution operation on the space ''Cc''(''G'') of complex-valued continuous functions on ''G'' with compact support; ''Cc''(''G'') can then be given any of various norms and the completion will be a group algebra. To define the convolution operation, let ''f'' and ''g'' be two functions in ''Cc''(''G''). For ''t'' in ''G'', define : * gt) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''-simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossed Product
In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups. (Roughly speaking, ''crossed product'' is the expected structure for a group ring of a semidirect product group. Therefore crossed products have a ring theory aspect also. This article concentrates on an important case, where they appear in functional analysis.) Motivation Recall that if we have two finite groups G and ''N'' with an action of ''G'' on ''N'' we can form the semidirect product N \rtimes G. This contains ''N'' as a normal subgroup, and the action of ''G'' on ''N'' is given by conjugation in the semidirect product. We can replace ''N'' by its complex group algebra ''C'' 'N'' and again form a product C \rtimes G in a similar way; this algebra is a sum of subspaces ''gC'' 'N''as ''g'' runs through ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand Representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-algebras, this representation is an isometric isomorphism. In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand–Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix. Historical remarks One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener (see the citation below), characterizing the elements of the group algebras ''L''1(R) and \ell^1() whose translates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
