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Multiplier Algebra
In mathematics, the multiplier algebra, denoted by ''M''(''A''), of a C*-algebra ''A'' is a unital C*-algebra that is the largest unital C*-algebra that contains ''A'' as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification. Multiplier algebras were introduced by . For example, if ''A'' is the C*-algebra of compact operators on a separable Hilbert space, ''M''(''A'') is ''B''(''H''), the C*-algebra of all bounded operators on ''H''. Definition An ideal ''I'' in a C*-algebra ''B'' is said to be essential if ''I'' ∩ ''J'' is non-trivial for every ideal ''J''. An ideal ''I'' is essential if and only if ''I''⊥, the "orthogonal complement" of ''I'' in the Hilbert C*-module ''B'' is . Let ''A'' be a C*-algebra. Its multiplier algebra ''M''(''A'') is any C*-algebra satisfying the following universal property: for any C*-algebra ''D'' containing ''A'' as an ideal, there exists a unique *-homomorphism φ: ''D'' → ''M''(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing set, absorbing Absolutely convex set, 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 number, complex numbers \Complex. A real-valued function p : X \to \R is called a if it satisfies the following two conditions: # Subadditive function, Subadditivity/Triangle inequality: p(x + y) \leq p(x) + p(y) for all x, y \in X. # Homogeneous function, 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 zer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corona Set
Corona (from the Latin for 'crown') most commonly refers to: * Stellar corona, the outer atmosphere of the Sun or another star * Corona (beer), a Mexican beer * Corona, informal term for the coronavirus or disease responsible for the COVID-19 pandemic: ** SARS-CoV-2, severe acute respiratory syndrome coronavirus 2 ** COVID-19, coronavirus disease 2019 Corona may also refer to: Architecture * Corona, a part of a cornice * The Corona, Canterbury Cathedral, the east end of Canterbury Cathedral Businesses and brands Food and drink * Corona (beer), a Mexican beer brand * Corona (restaurant), in the Netherlands * Corona (soft drink), a former brand Technology * Corona (software), a mobile app creation tool * Corona Data Systems, 1980s microcomputer supplier * Corona Labs Inc., an American software company * Corona Typewriter Company, merged into Smith Corona in 1926 * Corona, a version of Microsoft's Xbox 360 * Chaos Corona, rendering software Entertainment, ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calkin Algebra
In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of ''B''(''H''), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space ''H'', by the ideal ''K''(''H'') of compact operators. Here the addition in ''B''(''H'') is addition of operators and the multiplication in ''B''(''H'') is composition of operators; it is easy to verify that these operations make ''B''(''H'') into a ring. When scalar multiplication is also included, ''B''(''H'') becomes in fact an algebra over the same field over which ''H'' is a Hilbert space. Properties * Since ''K''(''H'') is a maximal norm-closed ideal in ''B''(''H''), the Calkin algebra is simple. In fact, ''K''(''H'') is the only closed ideal in ''B''(''H''). * As a quotient of a C*-algebra by a two-sided ideal, the Calkin algebra is a C*-algebra itself and there is a short exact sequence ::0 \to K(H) \to B(H) \to B(H)/K(H) \to 0 :which induces a six-term cyclic exact sequenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum Of A C*-algebra
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra ''A'', denoted ''Â'', is the set of unitary equivalence classes of irreducible *-representations of ''A''. A *-representation π of ''A'' on a Hilbert space ''H'' is irreducible if, and only if, there is no closed subspace ''K'' different from ''H'' and which is invariant under all operators π(''x'') with ''x'' ∈ ''A''. We implicitly assume that irreducible representation means ''non-null'' irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum ''Â'' is also naturally a topological space; this is similar to the notion of the spectrum of a ring. One of the most important applications of this concept is to provide a notion of dual object for any locally compact group. This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand–Naimark Theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra ''A'' is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra. Details The Gelfand–Naimark representation π is the Hilbert space analogue of the direct sum of representations π''f'' of ''A'' where ''f'' ranges over the set of pure states of A and π''f'' is the irreducible representation associated to ''f'' by the GNS construction. Thus the Gelfand–Naimark representation acts on the Hilbert direct sum of the Hilbert spaces ''H''''f'' by : \pi(x) bigoplus_ H_f= \bigoplus_ \pi_f(x)H_f. π(''x'') is a bounded linear operator since it is the direct sum o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vanish At Infinity
In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces. Aside from this difference, both of these notions correspond to the intuitive notion of adding a point at infinity, and requiring the values of the function to get arbitrarily close to zero as one approaches it. This definition can be formalized in many cases by adding an (actual) point at infinity. Definitions A function on a normed vector space is said to if the function approaches 0 as the input grows without bounds (that is, f(x) \to 0 as \, x\, \to \infty). Or, :\lim_ f(x) = \lim_ f(x) = 0 in the specific case of functions on the real line. For example, the function :f(x) = \frac defined on the real line vanishes at infinity. Alternatively, a function f on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topologies On The Set Of Operators On A Hilbert Space
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operators on the Banach space X. Consider the statement that (T_n)_ converges to some operator T on X. This could have several different meanings: * If \, T_n - T\, \to 0, that is, the operator norm of T_n - T (the supremum of \, T_n x - T x \, _X, where x ranges over the unit ball in X) converges to 0, we say that T_n \to T in the uniform operator topology. * If T_n x \to Tx for all x \in X, then we say T_n \to T in the strong operator topology. * Finally, suppose that for all x \in X we have T_n x \to Tx in the weak topology of X. This means that F(T_n x) \to F(T x) for all continuous linear functionals F on X. In this case we say that T_n \to T in the weak operator topology. List of topologies on B(''H'') There are many topologies that can be defined o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idealizer
In abstract algebra, the idealizer of a subsemigroup ''T'' of a semigroup ''S'' is the largest subsemigroup of ''S'' in which ''T'' is an Semigroup#Subsemigroups and ideals, ideal. Such an idealizer is given by :\mathbb_S(T)=\. In ring theory, if ''A'' is an additive subgroup of a ring (mathematics), ring ''R'', then \mathbb_R(A) (defined in the multiplicative semigroup of ''R'') is the largest subring of ''R'' in which ''A'' is a two-sided ideal. In Lie algebra, if ''L'' is a Lie ring (or Lie algebra) with Lie product [''x'',''y''], and ''S'' is an additive subgroup of ''L'', then the set :\ is classically called the normalizer of ''S'', however it is apparent that this set is actually the Lie ring equivalent of the idealizer. It is not necessary to specify that [''S'',''r''] ⊆ ''S'', because anticommutativity of the Lie product causes [''s'',''r''] = −[''r'',''s''] ∈ ''S''. The Lie "normalizer" of ''S'' is the largest subring of ''L'' in whic ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
