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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] |
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] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E-theory
In mathematics, ''KK''-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980. It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C*-algebras by Lawrence G. Brown, Ronald G. Douglas, and Peter Arthur Fillmore in 1977. In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of nuclear C*-algebras, as it was the key to the solutions of many problems in operator K-theory, such as, for instance, the mere calculation of ''K''-groups. Furthermore, it was essential in the development of the Baum–Connes conjecture and plays a crucial role in noncommutative topology. ''KK''-theory was followed by a series of similar bifunctor constructions such as the ''E''-theory and the bivariant periodic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nigel Higson
Nigel David Higson (born 1963) is a Canadian math professor at Pennsylvania State University who received the 1996 Coxeter–James Prize. His doctorate came from Dalhousie University in 1985, under the supervision of Peter Fillmore. He works in the fields of operator algebra and K-theory. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 2012 he was chosen as one of the inaugural Fellows of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... retrieved 2015-06-12. References < ...
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Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vanderbilt University. He was awarded the Fields Medal in 1982. Career Source: Academic career timeline: (1966–1970) – Bachelor's degree from the École Normale Supérieure (now part of Paris Sciences et Lettres University). (1973) – doctorate from Pierre and Marie Curie University, Paris, France (1970–1974) – appointment at the French National Centre for Scientific Research, Paris (1975) – Queen's University at Kingston, Ontario, Canada (1976–1980) – the University of Paris VI (1979 – present) – the Institute of Advanced Scientific Studies, Bures-sur-Yvette, France (1981–1984) – the French National Centre for Scientific Research, Paris (1984–2017) – the , Paris (2003–2011) – Vanderbilt University, Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
KK-theory
In mathematics, ''KK''-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980. It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C*-algebras by Lawrence G. Brown, Ronald G. Douglas, and Peter Arthur Fillmore in 1977. In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of nuclear C*-algebras, as it was the key to the solutions of many problems in operator K-theory, such as, for instance, the mere calculation of ''K''-groups. Furthermore, it was essential in the development of the Baum–Connes conjecture and plays a crucial role in noncommutative topology. ''KK''-theory was followed by a series of similar bifunctor constructions such as the ''E''-theory and the bivariant periodic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gennadi Kasparov
Gennadi ( gr, Γεννάδι) is a Greek village, seat of the municipal unit of South Rhodes, on the island of Rhodes, South Aegean region. In 2011 its population was 671. Overview The village is 64 km from the town of Rhodes and 27 km from ancient Lindos Lindos (; grc-gre, Λίνδος) is an archaeological site, a fishing village and a former municipality on the island of Rhodes, in the Dodecanese, Greece. Since the 2011 local government reform it is part of the municipality Rhodes, of which it ... and 65 km from the Airport of Rhodes. It is an agriculture place with a bit of tourism located on the south east side of Rhodes coast. References External links South Rhodes website Populated places in Rhodes {{SouthAegean-geo-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-homology
In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space. In terms of C^*-algebras, it classifies the Fredholm modules over an algebra. An operator homotopy between two Fredholm modules (\mathcal,F_0,\Gamma) and (\mathcal,F_1,\Gamma) is a norm continuous path of Fredholm modules, t \mapsto (\mathcal,F_t,\Gamma), t \in ,1 Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The K^0(A) group is the abelian group of equivalence classes In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ... of even Fredholm modules over A. The K^1(A) group is the abelian group of equivalence classes of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AF C*-algebra
In mathematics, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the ''K''0 functor whose range consists of ordered abelian groups with sufficiently nice order structure. The classification theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple amenable stably finite C*-algebras. Its proof divides into two parts. The invariant here is ''K''0 with its natural order structure; this is a functor. First, one proves ''existence'': a homomorphism between invariants must lift to a *-homomorphism of algebras. Second, one shows ''uniqueness'': the lift must be unique up to approximate unitary equivalence. Classification then follows from what is known as ''the inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George A
George may refer to: People * George (given name) * George (surname) * George (singer), American-Canadian singer George Nozuka, known by the mononym George * George Washington, First President of the United States * George W. Bush, 43rd President of the United States * George H. W. Bush, 41st President of the United States * George V, King of Great Britain, Ireland, the British Dominions and Emperor of India from 1910-1936 * George VI, King of Great Britain, Ireland, the British Dominions and Emperor of India from 1936-1952 * Prince George of Wales * George Papagheorghe also known as Jorge / GEØRGE * George, stage name of Giorgio Moroder * George Harrison, an English musician and singer-songwriter Places South Africa * George, Western Cape ** George Airport United States * George, Iowa * George, Missouri * George, Washington * George County, Mississippi * George Air Force Base, a former U.S. Air Force base located in California Characters * George (Peppa Pig), a 2-year-old ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essentially Normal Operator
Essence ( la, essentia) is a polysemic term, used in philosophy and theology as a designation for the property or set of properties that make an entity or substance what it fundamentally is, and which it has by necessity, and without which it loses its identity. Essence is contrasted with accident: a property that the entity or substance has contingently, without which the substance can still retain its identity. The concept originates rigorously with Aristotle (although it can also be found in Plato), who used the Greek expression ''to ti ên einai'' (τὸ τί ἦν εἶναι, literally meaning "the what it was to be" and corresponding to the scholastic term quiddity) or sometimes the shorter phrase ''to ti esti'' (τὸ τί ἐστι, literally meaning "the what it is" and corresponding to the scholastic term haecceity) for the same idea. This phrase presented such difficulties for its Latin translators that they coined the word ''essentia'' (English "essence") to repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
