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Gelfand Naimark Theorem
Gelfand is a surname meaning "elephant" in the Yiddish language, and may refer to: People * Alan Gelfand, the inventor of the ollie, a skateboarding move * Alan E. Gelfand, a statistician * Boris Gelfand, a chess grandmaster * Israel Gelfand, a mathematician, * Mikhail Gelfand, a molecular biologist and bioinformacisist, a grandson of Israel Gelfand * Vladimir Gelfand, a Soviet-Jewish writer Notions in mathematics Named after Israel Gelfand: * the Gelfand representation allows a complete characterization of commutative C*-algebras as algebras of continuous complex-valued functions * the Gelfand–Naimark–Segal construction * the Gelfand–Naimark theorem * the Gelfand–Mazur theorem In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states that a Banach algebra with unit over the complex numbers in which every nonzero element is invertible is isometrically isomorp ... * a Gelfand pair, a pair (''G'',''K'') c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elephant
Elephants are the largest living land animals. Three living species are currently recognised: the African bush elephant ('' Loxodonta africana''), the African forest elephant (''L. cyclotis''), and the Asian elephant ('' Elephas maximus''). They are the only surviving members of the family Elephantidae and the order Proboscidea; extinct relatives include mammoths and mastodons. Distinctive features of elephants include a long proboscis called a trunk, tusks, large ear flaps, pillar-like legs, and tough but sensitive grey skin. The trunk is prehensile, bringing food and water to the mouth and grasping objects. Tusks, which are derived from the incisor teeth, serve both as weapons and as tools for moving objects and digging. The large ear flaps assist in maintaining a constant body temperature as well as in communication. African elephants have larger ears and concave backs, whereas Asian elephants have smaller ears and convex or level backs. Elephants are scatter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand–Mazur Theorem
In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states that a Banach algebra with unit over the complex numbers in which every nonzero element is invertible is isometrically isomorphic to the complex numbers, i. e., the only complex Banach algebra that is a division algebra is the complex numbers \mathbb. The theorem follows from the fact that the spectrum of any element of a complex Banach algebra is nonempty: for every element a of a complex Banach algebra A there is some complex number \lambda such that \lambda 1 - a is not invertible. This is a consequence of the complex-analyticity of the resolvent function. By assumption, \lambda 1 - a = 0. So a = \lambda \cdot 1. This gives an isomorphism from A to \mathbb. The theorem can be strengthened to the claim that there are (up to isomorphism) exactly three real Banach division algebras: the field of reals \mathbb, the field of complex numbers \mathbb, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russian-language Surnames
Russian is an East Slavic language belonging to the Balto-Slavic branch of the Indo-European language family. It is one of the four extant East Slavic languages, and is the native language of the Russians. It was the ''de facto'' and ''de jure'' official language of the former Soviet Union. Constitution and Fundamental Law of the Union of Soviet Socialist Republics, 1977: Section II, Chapter 6, Article 36 Russian has remained an official language of the Russian Federation, Belarus, Kazakhstan, Kyrgyzstan, and Tajikistan, and is still commonly used as a lingua franca in Ukraine, Moldova, the Caucasus, Central Asia, and to a lesser extent in the Baltic states and Israel. Russian has over 253 million total speakers worldwide. It is the most spoken native language in Europe, the most spoken Slavic language, as well as the most geographically widespread language of Eurasia. It is the world's seventh-most spoken language by number of native speakers, and the world's ninth-most ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germanic-language Surnames
The Germanic languages are a branch of the Indo-European language family spoken natively by a population of about 515 million people mainly in Europe, North America, Oceania, and Southern Africa. The most widely spoken Germanic language, English, is also the world's most widely spoken language with an estimated 2 billion speakers. All Germanic languages are derived from Proto-Germanic, spoken in Iron Age Scandinavia, Iron Age Northern Germany and along the North Sea and Baltic coasts. The West Germanic languages include the three most widely spoken Germanic languages: English with around 360–400 million native speakers; German, with over 100 million native speakers; and Dutch, with 24 million native speakers. Other West Germanic languages include Afrikaans, an offshoot of Dutch originating from the Afrikaners of South Africa, with over 7.1 million native speakers; Low German, considered a separate collection of unstandardized dialects, with roughly 4.35–7.15 millio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helfant
Helfant is a surname meaning "elephant" in the Yiddish language. Notable people with the surname include: * Adam Helfant, sports executive * Edwin Helfant Edwin Helfant (12 April 1926 Margate City, New Jersey – 15 February 1978 Ducktown, Atlantic City) was an American lawyer in Atlantic City, New Jersey. Edwin was a New Jersey Bar Association-certified lawyer and part-time Judiciary of New ... (1926–1978), American lawyer See also * Gelfond * Gelfand * Helfand {{surname Yiddish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helfand
Helfand is a surname meaning "elephant" in Yiddish. Notable people with the surname include: * David Helfand, American astronomer * Jessica Helfand, designer, author, and educator * Lev Borisovich Helfand, Soviet diplomat See also * Gelfand * Gelfond * Helfant Helfant is a surname meaning "elephant" in the Yiddish language. Notable people with the surname include: * Adam Helfant, sports executive * Edwin Helfant Edwin Helfant (12 April 1926 Margate City, New Jersey – 15 February 1978 Ducktown ... {{surname, Helfand Surnames of Jewish origin Yiddish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfond
Gelfand is a surname meaning "elephant" in the Yiddish language. Notable people with the surname include: * Alexander Gelfond (1906–1968), Soviet mathematician * Michael Gelfond, American computer scientist See also * Gelfand * Helfand * Helfant Helfant is a surname meaning "elephant" in the Yiddish language. Notable people with the surname include: * Adam Helfant, sports executive * Edwin Helfant Edwin Helfant (12 April 1926 Margate City, New Jersey – 15 February 1978 Ducktown ... {{surname Yiddish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand Triple
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place. Using this notion, a version of the spectral theorem for unbounded operators on Hilbert space can be formulated. "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the Dirac formulation of quantum mechanics." Motivation A function such as x \mapsto e^ , is an eigenfunction of the differential operator -i\frac on the real line , but isn't square-integrable for the usual (Lebesgue) measure on . To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the Hilbert space theory. This was supplied by the apparatus of distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand Pair
In mathematics, a Gelfand pair is a pair (''G'', ''K'') consisting of a group ''G'' and a subgroup ''K'' (called an Euler subgroup of ''G'') that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory. When ''G'' is a finite group, the simplest definition is, roughly speaking, that the (''K'', ''K'')-double cosets in ''G'' commute. More precisely, the Hecke algebra, the algebra of functions on ''G'' that are invariant under translation on either side by ''K'', should be commutative for the convolution on ''G''. In general, the definition of Gelfand pair is roughly that the restriction to ''K'' of any irreducible representation of ''G'' cont ... [...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] |
Yiddish Language
Yiddish, historically Judeo-German, is a West Germanic language historically spoken by Ashkenazi Jews. It originated in 9th-century Central Europe, and provided the nascent Ashkenazi community with a vernacular based on High German fused with many elements taken from Hebrew language, Hebrew (notably Mishnaic Hebrew, Mishnaic) and to some extent Aramaic. Most varieties of Yiddish include elements of Slavic languages and the vocabulary contains traces of Romance languages.Aram Yardumian"A Tale of Two Hypotheses: Genetics and the Ethnogenesis of Ashkenazi Jewry".University of Pennsylvania. 2013. Yiddish has traditionally been written using the Hebrew alphabet. Prior to World War II, there were 11–13 million speakers. 85% of the approximately 6 million Jews who were murdered in the Holocaust were Yiddish speakers,Solomon Birnbaum, ''Grammatik der jiddischen Sprache'' (4., erg. Aufl., Hamburg: Buske, 1984), p. 3. leading to a massive decline in the use of the language. Jewish ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand–Naimark–Segal Construction
In functional analysis, a discipline within mathematics, given a C^*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A (called ''states''). The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal. States and representations A *-representation of a C^*-algebra A on a Hilbert space H is a mapping \pi from A into the algebra of bounded operators on H such that * \pi is a ring homomorphism which carries involution on A into involution on operators * \pi is nondegenerate, that is the space of vectors \pi (x) \xi is dense as x ranges through A and \xi ranges through H. Note that if A has an identity, nondegeneracy means exactly \pi is unit-preserving, i.e. \pi maps the identity of A to the identity operator on H. A state on a C^*-algebra A is a positive linear functional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
