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Topic: Gelfand isomorphism

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	Gelfand representation - Wikipedia, the free encyclopedia
		The Gelfand representation theorem is one avenue in the development of spectral theory for normal operators.
		The Banach-Alaoglu theorem of functional analysis asserts that the unit ball the dual of a Banach space is weak-* compact.
		By the Gelfand isomorphism applied to C(x) this is -isomorphic to an algebra of continuous functions on a locally compact space.
	en.wikipedia.org /wiki/Gelfand_representation (919 words)

	Learn more about Category theory in the online encyclopedia. (Site not responding. Last check: 2007-10-15)
		Furthermore, different such constructions are often "naturally related" which leads to the concept of natural transformation, a way to "map" one functor to another.
		Throughout mathematics, one encounters "natural isomorphisms", things that are (essentially) the same in a "canonical way".
		A groupoid is a category in which every morphism is an isomorphism.
	www.onlineencyclopedia.org /c/ca/category_theory.html (2963 words)

	Gelfand representation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-15)
		The Gelfand representation theorem is one avenue in the development of (Click link for more info and facts about spectral theory) spectral theory for (Click link for more info and facts about normal operator) normal operators.
		The Gelfand map give rise to a (Click link for more info and facts about contravariant) contravariant functor from the category of C*-algebras and morphisms into the category of locally compact Hausdorff spaces and continuous maps.
		The (Click link for more info and facts about Gelfand-Naimark theorem) Gelfand-Naimark theorem is a result for arbitrary (abstract) (Click link for more info and facts about noncommutative) noncommutative C-algebras A, which though not quite analogous to the Gelfand* representation, does provide a concrete representation of A as an algebra of operators.
	www.absoluteastronomy.com /encyclopedia/G/Ge/Gelfand_representation.htm (971 words)

	Gelfand isomorphism (Site not responding. Last check: 2007-10-15)
		The Gelfand map give rise to a contravariant functor from the categoryof C*-algebras and morphisms into the category of locally compact Hausdorff spaces and continuous maps.
		The Gelfand-Naimark theorem is a result forarbitrary (abstract) noncommutative C-algebras A, which though notquite analogous to the Gelfand* representation, does provide a concrete representation of A as an algebra ofoperators.
		By the Gelfand isomorphism applied to C(x) this is -isomorphicto an algebra of continuous functions on a locally compact space.
	www.therfcc.org /gelfand-isomorphism-358215.html (677 words)

	C-algebra -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-15)*
		A C-algebra A is a (Click link for more info and facts about Banach algebra) Banach algebra over the field of (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers, together with a map : A → A called involution.
		More generally, one can consider finite (A union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets) direct sums of matrix algebras.
		In the latter case, we can use the fact that the structure of these is completely determined by the (Click link for more info and facts about Gelfand isomorphism) Gelfand isomorphism.
	www.absoluteastronomy.com /encyclopedia/C/C/C-algebra.htm (1706 words)

	Normed division algebra (Site not responding. Last check: 2007-10-15)
		In all of the above cases, the norm is given by the absolute value.
		The only commutative Banach division algebras over the reals (up to isomorphism) are the reals themselves, and the complex numbers.
		It was proved by Gelfand in 1941, based on the 1938 work of Mazur mentioned above.
	www.theezine.net /n/normed-division-algebra.html (121 words)

	Kids.net.au - Encyclopedia Category theory - (Site not responding. Last check: 2007-10-15)
		The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η
		is an isomorphism for every object X in C.
		C such that FG is naturally isomorphic to I
	www.kids.net.au /encyclopedia-wiki/ca/Category_theory (2107 words)

	Gelfand isomorphism Definition / Gelfand isomorphism Research (Site not responding. Last check: 2007-10-15)
		One reason is that mathematical knowledge is revised and updated in a different way; though arguably founded on experiment in some manner, it is not comparable to the natural sciences in this respect.
		[click for more], the Gelfand representation in functional analysisFunctional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions.
		It has its historical roots in the study of transformations such as the Fourier transform and in the study of differential and integral equations.
	www.elresearch.com /Gelfand_isomorphism (244 words)

	PlanetMath: derived category (Site not responding. Last check: 2007-10-15)
		Call a morphism of chain complexes a quasi-isomorphism if it induces an isomorphism on homology groups of the
		For example, any chain homotopy is a quasi-isomorphism, but not conversely.
		Cross-references: derived functors, global sections, fixed, functors, algebra, place, localization, inverse, complexes, groups, homology, isomorphism, maps, classes, chain homotopy, morphisms, chain complexes, category, abelian category
	planetmath.org /encyclopedia/DerivedCategory.html (181 words)

	[No title]
		The main theorem in this paper is to show that there exists an isomorphism between the symplectic Floer cohomology of a based loop group and the semi-infinite cohomology of the corresponding loop algebra.
		Our isomorphism in Theorem~\ref{main}, for some examples of infinite-dimensional Lie algebras, provides the geometric topology explanation of the semi-infinite cohomology, and rigorously get into the string theory through the isomorphism for the symplectic Floer cohomology (see \cite{wi1, wi2}).
		The space $\Lam^{\infty/2+}{\fg}^$ is isomorphic to the exterior algebra $\Lam^{\bullet} {\fg}^*$ as a module of the Clifford algebra $C^{\infty} {\CG}$.
	www.math.okstate.edu /~wli/research/a6.tex (5623 words)

	[No title]
		For a commutative unital \c algebra $B$, there is an isomorphism ${ }^\wedge$, called the Gelfand map, from $B$ onto $C(\p B)$, such that $\widehat b(P)=b+P$ (since the algebra is commutative, primitive ideals are maximal, so $b+P$ is scalar) [5, p.41].
		To prove that the ideal centre is isomorphic to $C_b(\p A)$, Dixmier takes an element $z$ of the ideal centre, and argues that for each irreducible representation $\pi$ of $A$, $\pi(z)$ (makes sense and) is a scalar.
		Since $ZM(A)$ is a commutative \c algebra, the Gelfand map is an isomorphism from $ZM(A)$ onto $C(\p ZM(A))$, so it is enough to show that $\r_{\id}^*$ is onto.
	www.austms.org.au /Gazette/1996/Apr96/ideals.tex (1062 words)

	c star algebra (Site not responding. Last check: 2007-10-15)
		This vector uniquely determines the isomorphism class of a finite dimensional C*-algebra.
		The Gelfand representation states that every commutative C-algebra is -isomorphic to an algebra of the form C
		In this case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism theorem or we can use the continuous functional calculus.
	www.yourencyclopedia.net /c_star_algebra.html (881 words)

	Re: unitary equivalence for C*-algebras
		There is a problem that there may be several different K so that your algebra is isomo]rphic to a subalgebra of C(K).
		The most natural K to try is that provided by the Gelfand theory since it is defined in terms of multiplicative linear functionals.
		For the case of the measure algebra, the obvious candidate would be the integers using the Fourier transform for the *-isomorphism.
	www.lns.cornell.edu /spr/2003-08/msg0052875.html (388 words)

	[No title]
		We recall that the Gelfand spectrum of a commutative $C^{*}$ algebra $\B$ (with identity) is the space $\M$ of multiplicative linear functionals $M : \B \to \complessi \, $, with the weak topology defined by $\B$ on $\M$.
		By Gelfand's theorem, $\B$ is isomorphic to the algebra $C(\M)$ of the continuous functions over its spectrum, with the Sup norm \citaref{M. Naimark, {\it Normed Rings\/}, Nordhoff, 1964; M. Takesaki, {\it Theory of Operator Algebras I\/}, Springer Verlag 1979}.
		A (generalized) probabilistic interpretation, (as a measure on the Gelfand spectrum of $\A_n$) is therefore allowed, {\it only for\/} $\theta = 0$, for the winding number in infinite volume, as a variable \lq\lq uniformly distributed over the integers\rq\rq.
	www.ma.utexas.edu /mp_arc/e/95-121.tex (2279 words)

	Geometry and Combinatorics in Arizona (Site not responding. Last check: 2007-10-15)
		A lot of inspiration and intuition for our abelianization construction is drawn from studying the permutation action of the symmetric group $S_n$ on real n-dimensional vector space in detail.
		The Bernstein-Gelfand-Gelfand correspondence is an isomorphism between the category of linear free complexes over the exterior algebra E and the category of graded free modules over the symmetric algebra S. I'll give a concrete introduction to the BGG correspondence, discussing several examples in detail and showing how to compute using Macaulay2.
		Then we'll apply BGG to study the Chen ranks conjecture of Cohen-Suciu, which gives values for the Chen ranks of the fundamental group G of an arrangement complement in terms of the resonance variety R^1(A).
	www.math.ethz.ch /~feichtne/Phx/fl_program.html (625 words)

	Citations: Quantum Schubert polynomials - Fomin, Gelfand, Postnikov (ResearchIndex)
		Similar calculations lead one to believe that (26) Sw (c; c) 0 for w 6= 1: In a more recent work [B F] formulas for the general degeneracy loci described in [A DF K] and [L M] are found by different methods.
		The generating operators of Q Schur functions and their interaction with divided differences First of all we recall the construction of Q Schur functions and their operator representation, for which full details and proofs may be found in the book [HH] of Hoffman and Humphreys.
		Towards the end of the paper, we construct the quantum analogue of this algebra, and conjecture that its commutative subalgebra generated by the Dunkl elements is canonically isomorphic to the quantum cohomology ring of the flag manifold.
	citeseer.ist.psu.edu /context/51012/559202 (3525 words)

	Dictionary of Meaning www.mauspfeil.net
		The ''Gelfand map'' on the commutative C*-algebra ''A'' is defined as follows: :
		The Gelfand-Naimark theorem is a result for arbitrary (abstract) commutative noncommutative C-algebras ''A'', which though not quite analogous to the Gelfand* representation, does provide a concrete representation of ''A'' as an algebra of operators.
		There you find a list of all editors and the possibility to edit the original text of the article Gelfand representation.
	www.mauspfeil.net /Gelfand_representation.html (830 words)

	Citations: Schubert cells and the cohomology of the spaces G=P - Gelfan'd, Gelfan'd (ResearchIndex)
		The cohomology of the second space is by definition the B equivariant cohomology H B (G=B) Thus, H (BB Theta BG BB) H B (G=B) Under this isomorphism, the class of the diagonal corresponds to the B equivariant fundamental class of a point.
		In and [8] the problem of expressing the (Poincar e dual to) the class of a point in G=B in terms of the isomorphism H (G=B) R=J is solved (here J is the ideal in R generated by positive degree elements in S) Theorem 1.
		Under this isomorphism the Poincar e dual of a point in G=B is represented by jW j Gamma1 Q ff 0 ff mod J (8, x4.5] 3, Theorem 3.
	citeseer.ist.psu.edu /context/26088/0 (3449 words)

	Encyclopedia: Gelfand-Naimark theorem (Site not responding. Last check: 2007-10-15)
		The Gelfand representation or Gelfand isomorphism for a commutative C*-algebra with unit
		Gelfand and M. Naimark, On the imbeding of normed rings into the ring of operators on aHilbert space, Math.
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/Gelfand_Naimark-theorem (545 words)

	Atlas: On Gelfand's Problem concerning graphs, lattices, and ultrametric spaces by Alex J. Lemin
		A few yeas ago Professor Israel Gelfand set a Problem to describe all finite ultrametric spaces up to isometry using Graph Theory language (recall that (X, d) is ultrametric if it satisfies the strong triangle inequality d(x, z) <= max[d(x, y), d(y, z)]).
		We give a solution using the Theorem on isomorphism between the category of ultrametric spaces and a category LAT* of complete, atomic, tree-like and real graduated lattices (see [1]).
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagl-22.
	atlas-conferences.com /c/a/g/l/22.htm (470 words)

	[No title]
		The Gelfand topology of $\Delta$ is the weak topology induced by $\hat{A}$, i.e., the weakest topology that makes every $\hat{x}$ continuous.
		Since there is a one-to-one correspondence between the maximal ideals of $A$ and the members of $\Delta$ (by Theorem \ref{rudin}), $\Delta$, equipped with its Gelfand topology, is usually called the maximal ideal space of $A$.
		\\ {\rm (b)} \> The Gelfand transform is a nontrivial homomorphism of $A$ onto a subalgebra $\hat{A}$ \\ \> of $C(\Delta)$, whose kernel is {\em rad}$A$.
	www.math.uab.edu /sakata/dissertation/chap3.tex (5884 words)

	Gelfand representation - Wikpedia (Site not responding. Last check: 2007-10-15)
		In the general case, removal of a single point from a compact Hausdorff space yields a locally compact compact Hausdorff space.
		The Gelfand map on the commutative C*-algebra A is defined as follows:
		This page was last modified 17:50, 22 Mar 2005.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Gelfand_isomorphism (761 words)

	C.J.Mulvey
		Considering the work of Pelletier and Rosicky on simple involutive quantales in the context of Gelfand quantales allows their characterisations to be relativised to this case, providing rather straightforward conditions for a Gelfand quantale to be simple.
		The spectrum is shown to be a compact, completely regular quantale, on which the Gelfand representation of the C-algebra in the C-algebra of continuous complex functions on the spectrum may be defined.
		A Gelfand representation theorem establishes that this is an isometric -isomorphism of C-algebras, in a way that generalises precisely the commutative case, placing into the context of quantales the insights of Giles and Kummer concerning the spectral representation of C*-algebras, as proposed in the paper and.
	www.maths.susx.ac.uk /Staff/CJM/research/CJMForthcomingPapers.htm (559 words)

	C-algebra biography .ms (Site not responding. Last check: 2007-10-15)*
		Around 1943, the work of Gel'fand, Mark Naimark and Irving Segal yielded an abstract characterisation of C*-algebras making no reference to operators.
		These properties can be established by use the continuous functional calculus or by reduction to commutative C*-algebras.
		In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism.
	www.biography.ms /C%2A_algebra.html (1287 words)

	Amazon.ca: Editorial Reviews Books: An Introduction to the Classification of Amenable C -Algebras (Site not responding. Last check: 2007-10-15)
		The theory and applications of C*-algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to non-commutative geometry and dynamical systems.
		By Gelfand transformation, the theory of C*-algebras is also regarded as non-commutative topology.
		About a decade ago, George A. Elliott initiated the program of classification of C-algebras (up to isomorphism*) by their K-theoretical data.
	www.amazon.ca /exec/obidos/tg/detail/-/books/9810246803/reviews (353 words)

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