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Singular Trace
In mathematics, a singular trace is a Von Neumann algebra#Weights, states, and traces, trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite-rank operator, finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of Lp space#The p-norm in countably infinite dimensions, square-summable sequences and spaces of Hilbert space#Examples, square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix ring, matrix algebras have no non-trivial singular traces and the Trace (linear algebra), matrix trace is the unique trace up to scaling. American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of Trace class, trace class operators. Therefore, in distincti ... [...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] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Functional
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Calkin Correspondence
In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence spaces (also called rearrangement invariant sequence spaces). The correspondence is implemented by mapping an operator to its singular value sequence. It originated from John von Neumann's study of symmetric norms on matrix algebras. It provides a fundamental classification and tool for the study of two-sided ideals of compact operators and their traces, by reducing problems about operator spaces to (more resolvable) problems on sequence spaces. Definitions A ''two-sided ideal'' ''J'' of the bounded linear operators ''B''(''H'') on a separable Hilbert space ''H'' is a linear subspace such that ''AB'' and ''BA'' belong to ''J'' for all operators ''A'' from ''J'' and ''B'' from ''B''(''H''). A sequence space ''j'' within ''l''∞ can be em ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncommutative Residue
In mathematics, noncommutative residue, defined independently by M. and , is a certain trace on the algebra of pseudodifferential operators on a compact differentiable manifold that is expressed via a local density. In the case of the circle, the noncommutative residue had been studied earlier by M. and Y. in the context of one-dimensional integrable systems. See also * Dixmier trace References * * * * * *{{Citation , last1=Wodzicki , first1=Mariusz , title=K-theory, arithmetic and geometry (Moscow, 1984--1986) , publisher=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 ... , location=Berlin, New York , series=Lecture Notes in Math. , doi= 10.1007/BFb0078372 , mr=923140 , year=1987 , volume=1289 , chapter=Noncommutative residue. I. Fundamentals , pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator Subspace
In mathematics, the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of operators in the ideal with bounded operators. Modern characterisation of the commutator subspace is through the Calkin correspondence and it involves the invariance of the Calkin sequence space of an operator ideal to taking Cesàro means. This explicit spectral characterisation reduces problems and questions about commutators and traces on two-sided ideals to (more resolvable) problems and conditions on sequence spaces. History Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured in the matrix mechanics, or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician Paul Halmos in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
Dixmier Trace
In mathematics, the Dixmier trace, introduced by , is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces. Some applications of Dixmier traces to noncommutative geometry are described in . Definition If ''H'' is a Hilbert space, then ''L''1,∞(''H'') is the space of compact linear operators ''T'' on ''H'' such that the norm :\, T\, _ = \sup_N\frac is finite, where the numbers ''μ''''i''(''T'') are the eigenvalues of , ''T'', arranged in decreasing order. Let :a_N = \frac. The Dixmier trace Tr''ω''(''T'') of ''T'' is defined for positive operators ''T'' of ''L''1,∞(''H'') to be :\operatorname_\omega(T)= \lim_\omega a_N where lim''ω'' is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties: *lim''ω''(''α''''n'') ≥ 0 if all ''α''''n'' ≥ 0 (positivi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Trace-class Operator
In mathematics, a weak trace class operator is a compact operator on a separable Hilbert space ''H'' with singular values the same order as the harmonic sequence. When the dimension of ''H'' is infinite, the ideal of weak trace-class operators is strictly larger than the ideal of trace class operators, and has fundamentally different properties. The usual operator trace on the trace-class operators does not extend to the weak trace class. Instead the ideal of weak trace-class operators admits an infinite number of linearly independent quasi-continuous traces, and it is the smallest two-sided ideal for which all traces on it are singular traces. Weak trace-class operators feature in the noncommutative geometry of French mathematician Alain Connes. Definition A compact operator ''A'' on an infinite dimensional separable Hilbert space ''H'' is ''weak trace class'' if μ(''n'',''A'') O(''n''−1), where μ(''A'') is the sequence of singular values. In mathematical notation t ... [...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] |
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