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Mathematical representation in functional analysis
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: a way of representing commutative
Gelfand_representation
Soviet mathematician (1913–2009)
theory; the Gelfand–Naimark theorem; the Gelfand–Naimark–Segal construction; Gelfand–Shilov spaces; the Gelfand–Pettis integral; the representation theory
Israel_Gelfand
In representation theory, a branch of mathematics, the Gelfand–Graev representation is a representation of a reductive group over a finite field introduced
Gelfand–Graev_representation
Correspondence in functional analysis
{\displaystyle *} -representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal. A ∗ {\displaystyle *} -representation of a C ∗
Gelfand–Naimark–Segal construction
Gelfand–Naimark–Segal_construction
Type of topological space in mathematics
homeomorphism) locally compact Hausdorff space X. This is shown using the Gelfand representation. The notion of local compactness is important in the study of topological
Locally_compact_space
Mathematics theorem in functional analysis
and πf is the irreducible representation associated to f by the GNS construction. Thus the Gelfand–Naimark representation acts on the Hilbert direct
Gelfand–Naimark_theorem
Mathematical object
definition of Gelfand pair is roughly that the restriction to K of any irreducible representation of G contains the trivial representation of K with multiplicity
Gelfand_pair
Proof that every structure with certain properties is isomorphic to another structure
The Gelfand–Naimark–Segal construction embeds any C*-algebra in an algebra of bounded operators on some Hilbert space. The Gelfand representation (also
Representation_theorem
Particular kind of algebraic structure
normal, the Gelfand representation is isometric; in particular, it is injective and its image is closed. But the image of the Gelfand representation is dense
Banach_algebra
Surname list
Gelfand representation allows a complete characterization of commutative C*-algebras as algebras of continuous complex-valued functions the Gelfand–Naimark–Segal
Gelfand
continuous functions, which determine the group completely. Gelfand–Naimark theorem Representation theory И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные
Gelfand–Raikov_theorem
Representation of the symmetry group of spacetime in special relativity
Representation theory. A first course, Graduate Texts in Mathematics, vol. 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249 Gelfand
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
one-dimensional subspaces. In this way Gelfand and Tsetlin were able to obtain a basis of any irreducible representation of U ( N ) {\displaystyle U(N)} or
Restricted_representation
Function in mathematical analysis
functions over a compact space, this turns it into a C* algebra (cf. Gelfand representation). The uniform metric between two bounded functions f , g : X → Y
Uniform_norm
Integral equation
off. This equation is derived from the Gelfand–Levitan integral equation, using the Povzner–Levitan representation. Suppose that for a potential u ( x )
Marchenko_equation
{\displaystyle W} – and commutes with W {\displaystyle W} as a G action. Gelfand pair Claudio Procesi (2007) Lie Groups: an approach through invariants
Hecke algebra of a finite group
Hecke_algebra_of_a_finite_group
viewed as noncommutative generalizations of probability measures. By Gelfand representation, every commutative C*-algebra A is of the form C0(X) for some locally
State_(functional_analysis)
Gelfand, major contributor to numerous areas of mathematics, including group theory, representation theory and linear algebra, author of the Gelfand representation
List_of_Russian_scientists
Topological complex vector space
f K } {\displaystyle \{f_{K}\}} is an approximate identity. The Gelfand representation states that every commutative C*-algebra is *-isomorphic to the
C*-algebra
algebra reflexive operator algebra Calkin algebra Gelfand representation Gelfand–Naimark theorem Gelfand–Naimark–Segal construction Von Neumann algebra Abelian
List of functional analysis topics
List_of_functional_analysis_topics
Algebraic structure
spectrum of a linear operator; see Spectrum of a C*-algebra and Gelfand representation. Matsumura 1989, p. 143, §7, Remarks Matsumura 1989, §19, Theorem
Commutative_ring
Gelfand, major contributor to numerous areas of mathematics, including group theory, representation theory and linear algebra, author of the Gelfand representation
List of Russian mathematicians
List_of_Russian_mathematicians
Branch of mathematics
determines X {\displaystyle X} up to homeomorphism, by the commutative Gelfand representation. Similarly, the category of affine schemes in algebraic geometry
Noncommutative_geometry
Collection of mathematical theories
address Banach algebras in general. This development leads to the Gelfand representation, which covers the commutative case, and further into non-commutative
Spectral_theory
Unitary representations of a Lie group
irreducible unitary representations of the Lie group SL(2, R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952). We choose
Representation theory of SL2(R)
Representation_theory_of_SL2(R)
{A}}=C^{*}(a,e)} . The actual construction is almost immediate from the Gelfand representation: it suffices to assume A {\displaystyle {\mathcal {A}}} is the C*-algebra
Continuous functional calculus
Continuous_functional_calculus
Construction for adding objects to a Hilbert space
In mathematics and physics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction which can enlarge
Rigged_Hilbert_space
Mathematical concept
X. This result follows from the spectral radius formula and the Gelfand representation. (Gamelin 2005, p. 25) Gamelin, Theodore W. (2005). Uniform Algebras
Uniform_algebra
Ukrainian mathematician (born 1961)
and S.A. Ovsienko, advanced the theory of Harish-Chandra subalgebras and Gelfand–Tsetlin modules. This research established a foundation for the study of
Vyacheslav_Futorny
Abstract mathematics relationship
maximal ideals. This is the Gelfand representation. In lattice theory, there are a number of dualities, based on representation theorems that connect certain
Equivalence_of_categories
Conjectures connecting number theory and geometry
philosophy of cusp forms formulated a few years earlier by Harish-Chandra and Gelfand, the work and Harish-Chandra's approach on semisimple Lie groups, and in
Langlands_program
Writing Lie algebra sets as matrices
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra
Lie_algebra_representation
unitary representation of G {\displaystyle G} in a natural way. Let Φ : G → L ( H ) {\displaystyle \Phi :G\to L(H)} be a unitary representation of G {\displaystyle
Positive-definite function on a group
Positive-definite_function_on_a_group
Israeli mathematician
works in algebraic geometry, representation theory, and number theory. Bernstein received his Ph.D. in 1972 under Israel Gelfand at Moscow State University
Joseph_Bernstein
analysis Tempered representation Positive definite function on a group Symmetric space Gelfand pair If σ is a unitary representation of G, then σ ( f )
Zonal_spherical_function
Israel Gelfand, contributed to many areas of mathematics, including group theory, representation theory and linear algebra, author of the Gelfand representation
List_of_Russian_people
Canadian-American mathematician of Greek origin and operations researcher (1914–1981)
from ZF without use of the Axiom of Choice. Banach–Alaoglu theorem Gelfand representation List of functional analysis topics Superabundant number – Article
Leonidas_Alaoglu
(1993), "A proof of Jantzen conjectures" (PDF), in Gelʹfand, Sergei; Gindikin, Simon (eds.), I. M. Gelʹfand Seminar, Adv. Soviet Math., vol. 16, Providence
Jantzen_filtration
Directed graph which is also a multigraph
Elements of the Representation Theory of Associative Algebras, Cambridge University Press, ISBN 978-0-521-88218-7 Bernšteĭn, I. N.; Gelʹfand, I. M.; Ponomarev
Quiver_(mathematics)
Russian mathematician (1922–2017)
Russian mathematician. He is known as one of the namesakes in the Gelfand–Graev representation. Graev received his doctorate in 1947 from Lomonosov Moscow State
Mark_Iosifovich_Graev
American mathematician
Sciences. The Gelfand–Serganova theorem gives a geometric characterization of Coxeter matroids; it was published by Serganova and Israel Gelfand in 1987 as
Vera_Serganova
related to Wiener's Tauberian theorem and its generalization the Gelfand representation to commutative Banach algebras, C*-algebras and locally compact
Jorge_M._López
Representation theory of the symmetries of manifolds
M. A survey paper from 1975 of the subject by Anatoly Vershik, Israel Gelfand and M. I. Graev attributes the original interest in the topic to research
Representation theory of diffeomorphism groups
Representation_theory_of_diffeomorphism_groups
Woman assigned male at birth
90 (8): 4836–4845. doi:10.1210/jc.2004-2063. PMID 15840738. Sherwin BB, Gelfand MM, Brender W (1985). "Androgen enhances sexual motivation in females:
Trans_woman
Russian-American mathematician
under the mentorship of Joseph Bernstein, Alexandre Kirillov and Israel Gelfand. He worked in the mathematical laboratory of Vladimir Keilis-Borok at the
Andrei_Zelevinsky
Springer-Verlag, ISBN 3-540-10103-9, MR 0607504 Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969), Representation theory and automorphic functions, Academic
Uniformly bounded representation
Uniformly_bounded_representation
Representation theory of the symplectic group
In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David
Oscillator_representation
Mathematical operation
In mathematics, the Zak transform (also known as the Gelfand mapping) is a certain operation which takes as input a function of one variable and produces
Zak_transform
enunciated by Israel Gelfand, and the philosophy is a precursor of the Langlands program. A consequence for thinking about representation theory is that cuspidal
Parabolic_induction
Mathematical conjectures in class field theory
translation in volume 2 of Gelfand's collected works. Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) [1966], Representation theory and automorphic
Local_Langlands_conjectures
Russian mathematician
known for her contributions to representation theory and functional analysis in her collaboration with Israel Gelfand, and the Shapiro-Lobatinski condition
Zorya_Shapiro
Soviet mathematician (1909–1978)
symmetric operators, and the representation theory of locally compact operators. His collaboration with Israel Gelfand in the 1930s and 1940s led to
Mark_Naimark
Soviet-Israeli mathematician
doctorate under Alexandre Kirillov in 1969 and was a member of Israel Gelfand's school of mathematics. He is Jewish, and emigrated from the Soviet Union
David_Kazhdan
Mathematical transform that expresses a function of time as a function of frequency
ISBN 978-3-540-59179-5 Gelfand, I.M.; Shilov, G.E. (1964), Generalized Functions, vol. 1, New York: Academic Press (translated from Russian) Gelfand, I.M.; Shilov
Fourier_transform
Russian mathematician (born 1962)
Determinants, by I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky". Mathematical Association of America. Retrieved 1 Jul 2020. Gelfand, Israel M.; Kapranov
Mikhail_Kapranov
Functions on special groups related to their matrix representations
Israel Gelfand realized that many classical special functions and orthogonal polynomials are expressible as the matrix coefficients of representation of Lie
Matrix_coefficient
following up on earlier investigations of Bernstein–Gelfand–Gelfand and Demazure in representation theory in the 1970s, Lascoux and Schützenberger in combinatorics
Schubert_variety
Normed vector space that is complete
{\displaystyle C(K)} in the smaller category of commutative C*-algebras. Gelfand's representation theorem for commutative C*-algebras states that every commutative
Banach_space
Concept in mathematics
non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group
Universal_enveloping_algebra
Various mathematical dualites
of Koszul duality was introduced by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand. It establishes a duality between the derived category of a symmetric
Koszul_duality
Quantity used to describe the mathematical state of a dynamical system
applies the Gelfand transform for Abelian C*-algebras to represent abstract observables as functions on a representation space of states (Gelfand space).
State_variable
finite-dimension. Gelfand 1. The Gelfand–Mazur theorem states that a Banach algebra that is a division ring is the field of complex numbers. 2. The Gelfand representation
Glossary of functional analysis
Glossary_of_functional_analysis
Russian mathematician
He continued to study mathematical physics under direction of Israel Gelfand. After Khrushchev's liberalization, he joined the Department of Mathematics
Felix_Berezin
Topic in abstract algebra
by the introduction of reflection functors by Joseph Bernšteĭn, Israel Gelfand, and V. A. Ponomarev (1973); these functors were used to relate representations
Tilting_theory
Russian American mathematician (born 1957)
Kirillov and Israel Gelfand. Ginzburg wrote a textbook Representation theory and complex geometry with Neil Chriss on geometric representation theory. A paper
Victor_Ginzburg
Integral polynomial
In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial P y , w ( q ) {\displaystyle P_{y,w}(q)} is a member of a family of integral
Kazhdan–Lusztig_polynomial
Soviet mathematician (1920–1991)
Integral Geometry and Representation Theory by I. M. Gelfand, M. I. Graev and N. Ya. Vilenkin, Academic Press, 1966 Representation of Lie Groups and Special
Naum_Yakovlevich_Vilenkin
American mathematician
shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career, Segal became known for his developments in quantum
Irving_Segal
Russian mathematician
Kirillov studied at Moscow State University where he was a student of Israel Gelfand. His Ph.D. (kandidat) dissertation Unitary representations of nilpotent
Alexandre_Kirillov
Subgroup of GLn(k)
a given non-zero vector in kn. Mirabolic subgroups were introduced by (Gelfand & Kajdan 1975). The image of a mirabolic subgroup in the projective general
Mirabolic_group
American mathematician
The unity of mathematics: in honor of the ninetieth birthday of I. M. Gelfand, Birkhäuser 2006 Editor with Shlomo Gelaki and Steven Shnider: Quantum
Pavel_Etingof
the cohomology of algebraic varieties. It was originally conjectured by Gelfand and MacPherson. The first case of the decomposition theorem arises via
Decomposition theorem of Beilinson, Bernstein and Deligne
Decomposition_theorem_of_Beilinson,_Bernstein_and_Deligne
Russian-American mathematician (born 1939)
of California, Davis. With Israel Gelfand he introduced in 1970 the Gelfand-Fuchs cohomology of Lie algebras. Gelfand-Fuchs cohomology has applications
Dmitry_Fuchs
Objects in representation theory of Lie algebras
Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used
Verma_module
Group of real 2×2 matrices with unit determinant
families of unitary representations, which were worked out in detail by Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952). Linear
SL2(R)
Russian mathematician (born 1953)
State University (MSU) under joint supervision of Dmitry Fuchs and Israel Gelfand. His diploma thesis was dedicated to characteristic classes of flags of
Boris_Feigin
Algebraic structure used in analysis
Gelfand–Fuks cohomology Hopf algebra Index of a Lie algebra Leibniz algebra Lie algebra cohomology Lie algebra extension Lie algebra representation Lie
Lie_algebra
Deformation of the group algebra of a Coxeter group
representations were studied by Ian G. Macdonald. More generally if (G, K) is a Gelfand pair then the resulting algebra turns out to be commutative. Example: If
Iwahori–Hecke_algebra
Sporadic simple group
history of the sporadic simple groups", in Corwin, L.; Gelfand, I. M.; Lepowsky, James (eds.), The Gelʹfand Mathematical Seminars, 1990–1992, Boston, MA: Birkhäuser
Baby_monster_group
Mathematical theorem
about the restriction of a representation of a group G to a subgroup H. In that context, the pair (G, H) is called a strong Gelfand pair. Let G be a reductive
Multiplicity-one_theorem
Differential calculus on function spaces
link] Gelfand & Fomin 2000, pp. 11–12, 99 Gelfand & Fomin 2000, p. 12, footnote 6 Gelfand & Fomin 2000, p. 8 Gelfand & Fomin 2000, p. 6 Gelfand & Fomin
Calculus_of_variations
Russian-American mathematician
Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996–1999, 85–108, Gelfand Math. Sem., Birkhäuser, Boston 2000; arXiv:math/9812158
Alexander_L._Rosenberg
algebras. In particular, the classical Gelfand–Tsetlin construction of a basis in the space of such a representation has a natural interpretation in the
Yangian
(introduced in the PhD thesis of Mikhail Kogan) which are special faces of the Gelfand-Tsetlin polytope. Schubert polynomials also can be written as a weighted
Schubert_polynomial
Classifies quivers of finite type in terms of Dynkin diagrams
representations to the roots of Kac–Moody algebras. Bernšteĭn, I. N.; Gelfand, I. M.; Ponomarev, V. A. (1973), "Coxeter functors, and Gabriel's theorem"
Gabriel's_theorem
Concept in mathematics
These examples were rediscovered and given new life in 1991 by Israel Gelfand and Vladimir Retakh. There, they develop quasideterminantal versions of
Quasideterminant
polytopes of toric varieties, several polytopes appearing in representation theory (such as the Gelfand–Zetlin polytopes and the string polytopes of Peter Littelmann
Newton–Okounkov_body
are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky
Generalized_Verma_module
In mathematics, representation of a reductive algebraic group
B, then a Whittaker model for a representation is an embedding of it into the induced (Gelfand–Graev) representation IndG U(χ), where χ is a non-degenerate
Whittaker_model
Russian-American mathematician
Oil and Gas. While a student there, he attended the seminar of Israel Gelfand and worked with Boris Feigin and Dmitry Fuchs. In 1989, upon receiving
Edward_Frenkel
Theorem of Fourier transforms of Borel measures
G {\displaystyle G} . This is in fact a one-to-one correspondence. The Gelfand–Fourier transform is an isomorphism between the group C*-algebra C ∗ (
Bochner's_theorem
Term in abstract algebra
algebra, the Birman–Murakami–Wenzl algebra, the blocks of the Bernstein–Gelfand–Gelfand category O {\displaystyle {\mathcal {O}}} of a semisimple Lie algebra
Cellular_algebra
Generalized function whose value is zero everywhere except at zero
Hörmander 1983, §6.1. Lange 2012, pp.29–30. Gelfand & Shilov 1966–1968, p. 212. Gelfand & Shilov 1966–1968, p. 26. Gelfand & Shilov 1966–1968, §2.1. Weisstein
Dirac_delta_function
Coefficients coupled with angular momentum
"Unitary-group approach to the many-electron correlation problem: Relation of Gelfand and Weyl tableau formulations". Phys. Rev. A. 14 (5): 1620. Bibcode:1976PhRvA
3-j_symbol
Universal construction of a complex Lie group from a real Lie group
2004, pp. 202–203 See: Bump 2004 Zhelobenko 1973 Zhelobenko 1973 See: Gelfand & Naimark 1950, section 18, for SL(n,C) Bruhat 1956, p. 187 for SO(n,C)
Complexification_(Lie_group)
Completes the Langlands program for general linear groups over algebraic function fields
ISSN 1618-1913, MR 0601520, S2CID 189769469 Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) [1966], Representation theory and automorphic functions
Lafforgue's_theorem
Duality between a group and its representations
C*-algebra of endomorphisms of the monoidal unit contains only scalars. Gelfand–Naimark theorem Doplicher, S.; Roberts, J. (1989). "A new duality theory
Tannaka–Krein_duality
Set of functions between two fixed sets
New York. p. 65. doi:10.1007/978-1-4757-4383-8. ISBN 978-1-4419-3092-7. Gelfand, I. M.; Fomin, S. V. (2000). Silverman, Richard A. (ed.). Calculus of variations
Function_space
Theorem
two completely positive maps each of which has a special form: A *-representation of A on some auxiliary Hilbert space K followed by An operator map of
Stinespring_dilation_theorem
Identifies the commutant of a specific von Neumann algebra
abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s
Commutation theorem for traces
Commutation_theorem_for_traces
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