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| 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 functional]]s on ''A'' (called ''states''). The correspondence is shown by an explicit construction of the *-representation from the state. The content of the GNS construction is contained in the second theorem below. It is named for [[Israel Gelfand]], [[Mark Naimark]], and [[Irving Segal]].
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| == States and representations ==
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| A '''*-representation''' of a [[C*-algebra]] ''A'' on a [[Hilbert space]] ''H'' is a [[map (mathematics)|map]]ping
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| π from ''A'' into the algebra of [[bounded operator]]s on ''H'' such that
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| * π is a [[ring homomorphism]] which carries [[Involution (mathematics)|involution]] on ''A'' into involution on operators
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| *π is [[nondegenerate]], that is the space of vectors π(''x'') ξ is dense as ''x'' ranges through ''A'' and ξ ranges through ''H''. Note that if ''A'' has an identity, nondegeneracy means exactly π is unit-preserving, i.e. π maps the identity of ''A'' to the identity operator on ''H''.
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| A [[state (functional analysis)|state]] on C*-algebra ''A'' is a [[positive linear functional]] ''f'' of norm 1. If ''A'' has a multiplicative unit element this condition is equivalent to ''f''(1) = 1.
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| For a representation π of a C*-algebra ''A'' on a Hilbert space ''H'', an element ξ is called a '''cyclic vector''' if the set of vectors
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| :<math>\{\pi(x)\xi:x\in A\}</math> | |
| is norm dense in ''H'', in which case π is called a '''cyclic representation'''. Any non-zero vector of an irreducible representation is cyclic. However, non-zero vectors in a cyclic representation may fail to be cyclic.
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| ''Note to reader:'' In our definition of inner product, the conjugate linear argument is the first argument and the linear argument is the second argument. This is done for reasons of compatibility with the physics literature. Thus the order of arguments in some of the constructions below is exactly the opposite from those in many mathematics textbooks.
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| Let π be a *-representation of a C*-algebra ''A'' on the Hilbert space ''H'' with cyclic vector ξ having norm 1. Then
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| :<math> x \mapsto \langle \xi, \pi(x)\xi\rangle </math>
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| is a state of ''A''. Given *-representations π, π' each with unit norm cyclic vectors ξ ∈ ''H'', ξ' ∈ ''K'' such that their respective associated states coincide, then π, π' are unitarily equivalent representations. The operator ''U'' that maps π(''a'')ξ to π'(''a'')ξ' implements the unitary equivalence.
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| The converse is also true. Every state on a C*-algebra is of the above type. This is the '''GNS construction''':
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| '''Theorem.''' Given a state ρ of ''A'', there is a *-representation π of ''A'' with distinguished cyclic vector ξ such that its associated state is ρ, i.e.
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| :<math>\rho(x)=\langle \xi, \pi(x) \xi \rangle</math>
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| for every ''x'' in ''A''. | |
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| The construction proceeds as follows: The algebra ''A'' acts on itself by left multiplication. Via ρ, one can introduce a Hilbert space structure on ''A'' compatible with this action.
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| Define on ''A'' a, possibly singular, [[inner product space|inner product]]
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| :<math> \langle x, y \rangle =\rho(x^*y).</math> | |
| Here singular means that the sesquilinear form may fail to satisfy the non-degeneracy property of inner product. By the [[Cauchy–Schwarz inequality]], the degenerate elements, ''x'' in ''A'' satisfying ρ(''x* x'')= 0, form a vector subspace ''I'' of ''A''. By a C*-algebraic argument, one can show that ''I'' is a [[left ideal]] of ''A''. The [[quotient space (linear algebra)|quotient space]] of the ''A'' by the vector subspace ''I'' is an inner product space. The [[Cauchy completion]] of ''A''/''I'' in the quotient norm is a Hilbert space ''H''.
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| One needs to check that the action π(''x'')''y'' = ''xy'' of ''A'' on itself passes through the above construction. As ''I'' is a left ideal of ''A'', π descends to the quotient space ''A''/''I''. The same argument showing ''I'' is a left ideal also implies that π(''x'') is a bounded operator on ''A''/''I'' and therefore can be extended uniquely to the completion. This proves the existence of a *-representation π.
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| If ''A'' has a multiplicative identity 1, then it is immediate that the equivalence class ξ in the GNS Hilbert space ''H'' containing 1 is a cyclic vector for the above representation. If ''A'' is non-unital, take an [[approximate identity]] {''e<sub>λ</sub>''} for ''A''. Since positive linear functionals are bounded, the equivalence classes of the net {''e<sub>λ</sub>''} converges to some vector ξ in ''H'', which is a cyclic vector for π.
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| It is clear that the state ρ can be recovered as a vector state on the GNS Hilbert space. This proves the theorem.
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| The above shows that there is a bijective correspondence between positive linear functionals and cyclic representations. Two cyclic representations π<sub>φ</sub> and π<sub>ψ</sub> with corresponding positive functionals φ and ψ are unitarily equivalent if and only if φ = ''α'' ψ for some positive number ''α''.
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| If ω, φ, and ψ are positive linear functionals with ω = φ + ψ, then π<sub>ω</sub> is unitarily equivalent to a subrepresentation of π<sub>φ</sub> ⊕ π<sub>ψ</sub>. The embedding map is given by
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| :<math>\pi_{\omega}(x) \xi_{\omega} \mapsto \pi_{\phi}(x) \xi_{\phi} \oplus \pi_{\psi}(x) \xi_{\psi}.</math>
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| The GNS construction is at the heart of the proof of the [[Gelfand–Naimark theorem]] characterizing C*-algebras as algebras of operators. A C*-algebra has sufficiently many pure states (see below) so that the direct sum of corresponding irreducible GNS representations is [[Faithful group action|faithful]].
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| The direct sum of the corresponding GNS representations of all positive linear functionals is called the '''universal representation''' of ''A''. Since every nondegenerate representation is a direct sum of cyclic representations, any other representation is a *-homomorphic image of π. <!-- Similarly, any other representation π' is [[quasi equivalent]] to a subrepresentation of π. -->
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| If π is the universal representation of a C*-algebra ''A'', the closure of π(''A'') in the weak operator topology is called the '''[[enveloping von Neumann algebra]]''' of ''A''. It can be identified with the double dual ''A**''.
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| == Irreducibility ==
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| Also of significance is the relation between [[irreducible (mathematics)|irreducible]] *-representations and extreme points of the convex set of states. A representation π on ''H'' is irreducible if and only if there are no closed subspaces of ''H'' which are invariant under all the operators π(''x'') other than ''H'' itself and the trivial subspace {0}.
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| '''Theorem'''. The set of states of a C*-algebra ''A'' with a unit element is a compact [[convex set]] under the weak-* topology. In general, (regardless of whether or not ''A'' has a unit element) the set of positive functionals of norm ≤ 1 is a compact convex set.
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| Both of these results follow immediately from the [[Banach–Alaoglu theorem]].
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| In the unital commutative case, for the C*-algebra ''C''(''X'') of continuous functions on some compact ''X'', [[Riesz–Markov–Kakutani representation theorem]] says that the positive functionals of norm ≤ 1 are precisely the Borel positive measures on ''X'' with total mass ≤ 1. It follows from [[Krein–Milman theorem]] that the extremal states are the Dirac point-mass measures.
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| On the other hand, a representation of ''C''(''X'') is irreducible if and only if it is one dimensional. Therefore the GNS representation of ''C''(''X'') corresponding to a measure μ is irreducible if and only if μ is an extremal state. This is in fact true for C*-algebras in general.
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| '''Theorem'''. Let ''A'' be a C*-algebra. If π is a *-representation of
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| ''A'' on the Hilbert space ''H'' with unit norm cyclic vector ξ, then
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| π is irreducible if and only if the corresponding state ''f'' is an [[extreme point]] of the convex set of positive linear functionals on ''A'' of norm ≤ 1.
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| To prove this result one notes first that a representation is irreducible if and only if the [[commutant]] of π(''A''), denoted by π(''A'')', consists of scalar multiples of the identity.
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| Any positive linear functionals ''g'' on ''A'' dominated by ''f'' is of the form
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| :<math> g(x^*x) = \langle \pi(x) \xi, \pi(x) T_g \, \xi \rangle </math>
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| for some positive operator ''T<sub>g</sub>'' in π(''A'')' with 0 ≤ ''T'' ≤ 1 in the operator order. This is a version of the [[Radon–Nikodym theorem]].
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| For such ''g'', one can write ''f'' as a sum of positive linear functionals: ''f'' = ''g'' + ''g' ''. So π is unitarily equivalent to a subrepresentation of π<sub>''g''</sub> ⊕ π<sub>''g' ''</sub>. This shows that π is irreducible if and only if any such π<sub>''g''</sub> is unitarily equivalent to π, i.e. ''g'' is a scalar multiple of ''f'', which proves the theorem.
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| Extremal states are usually called [[pure states]]. Note that a state is a pure state if and only if it is extremal in the convex set of states.
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| The theorems above for C*-algebras are valid more generally in the context of [[B-star algebra|B*-algebra]]s with approximate identity.
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| == Generalizations ==
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| The [[Stinespring factorization theorem]] characterizing [[completely positive map]]s is an important generalization of the GNS construction.
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| == History ==
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| Gelfand and Naimark's paper on the Gelfand–Naimark theorem was published in 1943.<ref>{{cite journal |author=[[I. M. Gelfand]], [[M. A. Naimark]] |title=On the imbedding of normed rings into the ring of operators on a Hilbert space |journal=[[Matematicheskii Sbornik]] |volume=12 |issue=2 |year=1943 |pages=197–217 |url=http://mi.mathnet.ru/eng/msb6155}} (also [http://www.google.com/books?id=DYCUp0JYU6sC&printsec=frontcover#PPA3,M1 Google Books], see pp. 3–20)</ref> Segal recognized the construction that was implicit in this work and presented it in sharpened form.<ref>[[Richard V. Kadison]]: ''Notes on the Gelfand–Neimark theorem''. In: Robert C. Doran (ed.): ''C*-Algebras: 1943–1993. A Fifty Year Celebration'', AMS special session commemorating the first fifty years of C*-algebra theory, January 13–14, 1993, San Antonio, Texas, American Mathematical Society, pp. 21–54, ISBN 0-8218-5175-6 ([http://www.google.com/books?id=DYCUp0JYU6sC&printsec=frontcover#PPA3,M1 available from Google Books], see pp. 21 ff.)</ref>
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| In his paper of 1947 Segal showed that it is sufficient, for any physical system that can be described by an algebra of operators on a Hilbert space, to consider the ''irreducible'' representations of a C*-algebra. In quantum theory this means that the C*-algebra is generated by the observables. This, as Segal pointed out, had been shown earlier by [[John von Neumann]] only for the specific case of the non-relativistic Schrödinger-Heisenberg theory.<ref>{{cite journal |author=[[I. E. Segal]]|title=Irreducible representations of operator algebras |journal=Bull. Am. Math. Soc. |volume=53 |issue= |year=1947 |pages=73–88 |url=http://www.ams.org/journals/bull/1947-53-02/S0002-9904-1947-08742-5/S0002-9904-1947-08742-5.pdf}}</ref>
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| ==References==
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| * [[William Arveson]], ''An Invitation to C*-Algebra'', Springer-Verlag, 1981
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| * [[Jacques Dixmier]], ''Les C*-algèbres et leurs Représentations'', Gauthier-Villars, 1969.<br/>English translation: {{cite book
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| | last =Dixmier
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| | first =Jacques
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| | authorlink =
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| | coauthors =
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| | title = C*-algebras
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| | publisher =North-Holland
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| | year = 1982
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| | location =
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| | pages =
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| | url =
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| | doi =
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| | id =
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| | isbn = 0-444-86391-5}}
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| * Thomas Timmermann, ''An invitation to quantum groups and duality: from Hopf algebras to multiplicative unitaries and beyond'', European Mathematical Society, 2008, ISBN 978-3-03719-043-2 – [http://books.google.com/books?id=S8sZiieo-04C&pg=PA371 Appendix 12.1, section: GNS construction (p. 371)]
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| * Stefan Waldmann: ''On the representation theory of [[deformation quantization]]'', In: ''Deformation Quantization: Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, May 31-June 2, 2001 (Studies in Generative Grammar) '', Gruyter, 2002, ISBN 978-3-11-017247-8, p. 107–134 – [http://books.google.com/books?id=xuq8CHNEFKoC&pg=PA113 section 4. The GNS construction (p. 113)]
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| ;Inline references:
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| {{reflist}}
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| {{DEFAULTSORT:Gelfand-Naimark-Segal construction}}
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| [[Category:Functional analysis]]
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| [[Category:C*-algebras]]
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| [[Category:Quantum field theory]]
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| [[ru:Алгебраическая квантовая теория]]
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