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77-399: A *-representation of a C*-algebra A on a Hilbert space H is a mapping π from A into the algebra of bounded operators on H such that A state on a 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. For a representation π of a C*-algebra A on a Hilbert space H , an element ξ

154-508: A n z n {\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}} where ∑ n = 0 ∞ | a n | 2 < ∞ . {\displaystyle \sum _{n=0}^{\infty }|a_{n}|^{2}<\infty \,.} Gelfand%E2%80%93Naimark theorem In mathematics , the Gelfand–Naimark theorem states that an arbitrary C*-algebra A

231-417: A ↦ ⟨ π ( a ) ξ , ξ ⟩ {\displaystyle a\mapsto \langle \pi (a)\xi ,\xi \rangle } is a state of A . Conversely, every state of A may be viewed as a vector state as above, under a suitable canonical representation. Theorem.  —  Given a state ρ of A , there is a *-representation π of A acting on

308-497: A ) , a , b ∈ A . {\displaystyle \langle a,b\rangle =\rho (b^{*}a),\;a,b\in A.} 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 λ } for A . Since positive linear functionals are bounded,

385-431: A ) = ⟨ π ( a ) ξ , ξ ⟩ {\displaystyle \rho (a)=\langle \pi (a)\xi ,\xi \rangle } as seen in the theorem below. Theorem.  —  Given a state ρ of A , let π, π' be *-representations of A on Hilbert spaces H , H ′ respectively each with unit norm cyclic vectors ξ ∈ H , ξ' ∈ H ′ such that ρ (

462-533: A ) = ⟨ π ( a ) ξ , ξ ⟩ = ⟨ π ′ ( a ) ξ ′ , ξ ′ ⟩ {\displaystyle \rho (a)=\langle \pi (a)\xi ,\xi \rangle =\langle \pi '(a)\xi ',\xi '\rangle } for all a ∈ A {\displaystyle a\in A} . Then π, π' are unitarily equivalent *-representations i.e. there

539-708: A set of measure zero . The inner product of functions f and g in L ( X , μ ) is then defined as ⟨ f , g ⟩ = ∫ X f ( t ) g ( t ) ¯ d μ ( t ) {\displaystyle \langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)} or ⟨ f , g ⟩ = ∫ X f ( t ) ¯ g ( t ) d μ ( t ) , {\displaystyle \langle f,g\rangle =\int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,} where

616-434: A Hilbert space H with distinguished unit cyclic vector ξ such that ρ ( a ) = ⟨ π ( a ) ξ , ξ ⟩ {\displaystyle \rho (a)=\langle \pi (a)\xi ,\xi \rangle } for every a in A . Define on A a semi-definite sesquilinear form ⟨ a , b ⟩ = ρ ( b ∗

693-577: A Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space . A Hilbert space is a special case of a Banach space . Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert , Erhard Schmidt , and Frigyes Riesz . They are indispensable tools in the theories of partial differential equations , quantum mechanics , Fourier analysis (which includes applications to signal processing and heat transfer ), and ergodic theory (which forms

770-548: A Hilbert space that, with the inner product induced by restriction , is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right. The sequence space l consists of all infinite sequences z = ( z 1 , z 2 , ...) of complex numbers such that the following series converges : ∑ n = 1 ∞ | z n | 2 {\displaystyle \sum _{n=1}^{\infty }|z_{n}|^{2}} The inner product on l

847-592: A definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory. Lebesgue spaces are function spaces associated to measure spaces ( X , M , μ ) , where X

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924-507: A distance function defined in this way, any inner product space is a metric space , and sometimes is known as a pre-Hilbert space . Any pre-Hilbert space that is additionally also a complete space is a Hilbert space. The completeness of H is expressed using a form of the Cauchy criterion for sequences in H : a pre-Hilbert space H is complete if every Cauchy sequence converges with respect to this norm to an element in

1001-548: A family of operators, each one having norm ≤ || x ||. Theorem . The Gelfand–Naimark representation of a C*-algebra is an isometric *-representation. It suffices to show the map π is injective , since for *-morphisms of C*-algebras injective implies isometric. Let x be a non-zero element of A . By the Krein extension theorem for positive linear functionals , there is a state f on A such that f ( z ) ≥ 0 for all non-negative z in A and f (− x * x ) < 0. Consider

1078-977: A non-negative integer and Ω ⊂ R , the Sobolev space H (Ω) contains L functions whose weak derivatives of order up to s are also L . The inner product in H (Ω) is ⟨ f , g ⟩ = ∫ Ω f ( x ) g ¯ ( x ) d x + ∫ Ω D f ( x ) ⋅ D g ¯ ( x ) d x + ⋯ + ∫ Ω D s f ( x ) ⋅ D s g ¯ ( x ) d x {\displaystyle \langle f,g\rangle =\int _{\Omega }f(x){\bar {g}}(x)\,\mathrm {d} x+\int _{\Omega }Df(x)\cdot D{\bar {g}}(x)\,\mathrm {d} x+\cdots +\int _{\Omega }D^{s}f(x)\cdot D^{s}{\bar {g}}(x)\,\mathrm {d} x} where

1155-408: A physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued work with Eugene Wigner . The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups. The significance of

1232-730: A real number x ⋅ y . If x and y are represented in Cartesian coordinates , then the dot product is defined by ( x 1 x 2 x 3 ) ⋅ ( y 1 y 2 y 3 ) = x 1 y 1 + x 2 y 2 + x 3 y 3 . {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.} The dot product satisfies

1309-579: A series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as  N → ∞ . {\displaystyle {\Biggl \|}\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}{\Biggr \|}\to 0\quad {\text{as }}N\to \infty \,.} This property expresses

1386-474: A special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces ) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations . They also form the basis of the theory of direct methods in the calculus of variations . For s

1463-512: A state is a pure state if and only if it is extremal in the convex set of states. The theorems above for C*-algebras are valid more generally in the context of B*-algebras with approximate identity. The Stinespring factorization theorem characterizing completely positive maps is an important generalization of the GNS construction. Gelfand and Naimark's paper on the Gelfand–Naimark theorem

1540-531: A suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness. The second development was the Lebesgue integral , an alternative to the Riemann integral introduced by Henri Lebesgue in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that

1617-636: Is countably infinite , it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable . The latter space is often in the older literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors , denoted by R , and equipped with the dot product . The dot product takes two vectors x and y , and produces

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1694-509: Is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. To say that a complex vector space H is a complex inner product space means that there is an inner product ⟨ x , y ⟩ {\displaystyle \langle x,y\rangle } associating a complex number to each pair of elements x , y {\displaystyle x,y} of H that satisfies

1771-412: Is a *-representation of A on the Hilbert space H with unit norm cyclic vector ξ, then 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. 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

1848-584: Is a compact convex set. Both of these results follow immediately from the Banach–Alaoglu theorem . 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

1925-562: Is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: | z | = x 2 + y 2 . {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.} The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w : ⟨ z , w ⟩ = z w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This

2002-428: Is a distance function means firstly that it is symmetric in x {\displaystyle x} and y , {\displaystyle y,} secondly that the distance between x {\displaystyle x} and itself is zero, and otherwise the distance between x {\displaystyle x} and y {\displaystyle y} must be positive, and lastly that

2079-609: Is a set, M is a σ-algebra of subsets of X , and μ is a countably additive measure on M . Let L ( X , μ ) be the space of those complex-valued measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function f in L ( X , μ ) , ∫ X | f | 2 d μ < ∞ , {\displaystyle \int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,} and where functions are identified if and only if they differ only on

2156-403: Is a unitary operator U from H to H ′ such that π'( a ) = Uπ( a )U* for all a in A . The operator U that implements the unitary equivalence maps π( a )ξ to π'( a )ξ' for all a in A . 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

2233-400: Is called a cyclic vector if the set of vectors 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 general cyclic representation may fail to be cyclic. Let π be a *-representation of a C*-algebra A on the Hilbert space H and ξ be a unit norm cyclic vector for π. Then

2310-486: Is called the weighted L space L w ([0, 1]) , and w is called the weight function. The inner product is defined by ⟨ f , g ⟩ = ∫ 0 1 f ( t ) g ( t ) ¯ w ( t ) d t . {\displaystyle \langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.} The weighted space L w ([0, 1])

2387-627: Is complex-valued. The real part of ⟨ z , w ⟩ gives the usual two-dimensional Euclidean dot product . A second example is the space C whose elements are pairs of complex numbers z = ( z 1 , z 2 ) . Then an inner product of z with another such vector w = ( w 1 , w 2 ) is given by ⟨ z , w ⟩ = z 1 w 1 ¯ + z 2 w 2 ¯ . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.} The real part of ⟨ z , w ⟩

Gelfand–Naimark–Segal construction - Misplaced Pages Continue

2464-433: Is defined by ‖ f ‖ 2 = lim r → 1 M r ( f ) . {\displaystyle \left\|f\right\|_{2}=\lim _{r\to 1}{\sqrt {M_{r}(f)}}\,.} Hardy spaces in the disc are related to Fourier series. A function f is in H ( U ) if and only if f ( z ) = ∑ n = 0 ∞

2541-457: Is defined by: ⟨ z , w ⟩ = ∑ n = 1 ∞ z n w n ¯ , {\displaystyle \langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,} This second series converges as a consequence of the Cauchy–Schwarz inequality and

2618-564: Is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and ( H , H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,H,\langle \cdot ,\cdot \rangle )} will form a dual system . The norm is the real-valued function ‖ x ‖ = ⟨ x , x ⟩ , {\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}\,,} and

2695-590: Is identical with the Hilbert space L ([0, 1], μ ) where the measure μ of a Lebesgue-measurable set A is defined by μ ( A ) = ∫ A w ( t ) d t . {\displaystyle \mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.} Weighted L spaces like this are frequently used to study orthogonal polynomials , because different families of orthogonal polynomials are orthogonal with respect to different weighting functions. Sobolev spaces , denoted by H or W , are Hilbert spaces. These are

2772-425: 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}. 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

2849-447: 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 . The Gelfand–Naimark representation π

2926-440: Is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg 's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras . In the 1940s, Israel Gelfand , Mark Naimark and Irving Segal gave

3003-517: Is related to both the length (or norm ) of a vector, denoted ‖ x ‖ , and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos ⁡ θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.} Multivariable calculus in Euclidean space relies on

3080-456: 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 π( x ) is a bounded linear operator since it is the direct sum of

3157-527: Is the Laplacian and (1 − Δ) is understood in terms of the spectral mapping theorem . Apart from providing a workable definition of Sobolev spaces for non-integer s , this definition also has particularly desirable properties under the Fourier transform that make it ideal for the study of pseudodifferential operators . Using these methods on a compact Riemannian manifold , one can obtain for instance

Gelfand–Naimark–Segal construction - Misplaced Pages Continue

3234-414: Is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z ⟩ = ⟨ z , w ⟩ ¯ . {\displaystyle \langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.} A Hilbert space

3311-499: The Banach *-algebra A having an approximate identity: It follows that an equivalent form for the C* norm on A is to take the above supremum over all states. The universal construction is also used to define universal C*-algebras of isometries. Remark . The Gelfand representation or Gelfand isomorphism for a commutative C*-algebra with unit A {\displaystyle A}

3388-968: The Hodge decomposition , which is the basis of Hodge theory . The Hardy spaces are function spaces, arising in complex analysis and harmonic analysis , whose elements are certain holomorphic functions in a complex domain. Let U denote the unit disc in the complex plane. Then the Hardy space H ( U ) is defined as the space of holomorphic functions f on U such that the means M r ( f ) = 1 2 π ∫ 0 2 π | f ( r e i θ ) | 2 d θ {\displaystyle M_{r}(f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f{\bigl (}re^{i\theta }{\bigr )}\right|^{2}\,\mathrm {d} \theta } remain bounded for r < 1 . The norm on this Hardy space

3465-708: The Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if w is any positive measurable function, the space of all measurable functions f on the interval [0, 1] satisfying ∫ 0 1 | f ( t ) | 2 w ( t ) d t < ∞ {\displaystyle \int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty }

3542-486: The Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis , in analogy with Cartesian coordinates in classical geometry. When this basis

3619-552: The completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers . The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus | z | , which is defined as the square root of the product of z with its complex conjugate : | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy

3696-463: The quotient vector space A / I is an involutive algebra and the norm factors through a norm on A / I , which except for completeness, is a C* norm on A / I (these are sometimes called pre-C*-norms). Taking the completion of A / I relative to this pre-C*-norm produces a C*-algebra B . By the Krein–Milman theorem one can show without too much difficulty that for x an element of

3773-824: The triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs: d ( x , z ) ≤ d ( x , y ) + d ( y , z ) . {\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.} This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality , which asserts | ⟨ x , y ⟩ | ≤ ‖ x ‖ ‖ y ‖ {\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|\|y\|} with equality if and only if x {\displaystyle x} and y {\displaystyle y} are linearly dependent . With

3850-421: The GNS representation π f with cyclic vector ξ. Since it follows that π f (x) ≠ 0, so π (x) ≠ 0, so π is injective. The construction of Gelfand–Naimark representation depends only on the GNS construction and therefore it is meaningful for any Banach *-algebra A having an approximate identity . In general (when A is not a C*-algebra) it will not be a faithful representation . The closure of

3927-653: The ability to compute limits , and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n {\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}} consisting of vectors in R is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞ . {\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.} Just as with

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4004-499: The concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics . In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators , and measurements are orthogonal projections . The relation between quantum mechanical symmetries and unitary operators provided an impetus for

4081-440: The convergence of the previous series. Completeness of the space holds provided that whenever a series of elements from l converges absolutely (in norm), then it converges to an element of l . The proof is basic in mathematical analysis , and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space). Prior to

4158-551: The development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists . In particular, the idea of an abstract linear space (vector space) had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as real or complex numbers ) without necessarily identifying these elements with "geometric" vectors , such as position and momentum vectors in physical systems. Other objects studied by mathematicians at

4235-479: The development of the unitary representation theory of groups , initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space ( Koopman–von Neumann classical mechanics ) and that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory . The algebra of observables in quantum mechanics

4312-435: The direct sum of corresponding irreducible GNS representations is faithful . The direct sum of the corresponding GNS representations of all states is called the universal representation of A . The universal representation of A contains every cyclic representation. As every *-representation is a direct sum of cyclic representations, it follows that every *-representation of A is a direct summand of some sum of copies of

4389-471: The distance d {\displaystyle d} between two points x , y {\displaystyle x,y} in H is defined in terms of the norm by d ( x , y ) = ‖ x − y ‖ = ⟨ x − y , x − y ⟩ . {\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.} That this function

4466-846: The dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when s is not an integer. Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. If Ω is a suitable domain, then one can define the Sobolev space H (Ω) as the space of Bessel potentials ; roughly, H s ( Ω ) = { ( 1 − Δ ) − s / 2 f | f ∈ L 2 ( Ω ) } . {\displaystyle H^{s}(\Omega )=\left\{(1-\Delta )^{-s/2}f\mathrel {\Big |} f\in L^{2}(\Omega )\right\}\,.} Here Δ

4543-460: The early 20th century. For example, the Riesz representation theorem was independently established by Maurice Fréchet and Frigyes Riesz in 1907. John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators . Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from

4620-402: The equivalence classes of the net { e λ } converges to some vector ξ in H , which is a cyclic vector for π. The method used to produce a *-representation from a state of A in the proof of the above theorem is called the GNS construction . For a state of a C*-algebra A , the corresponding GNS representation is essentially uniquely determined by the condition, ρ (

4697-452: The extremal states are the Dirac point-mass measures. 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. Theorem  —  Let A be a C*-algebra. If π

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4774-628: The following properties: It follows from properties 1 and 2 that a complex inner product is antilinear , also called conjugate linear , in its second argument, meaning that ⟨ x , a y 1 + b y 2 ⟩ = a ¯ ⟨ x , y 1 ⟩ + b ¯ ⟨ x , y 2 ⟩ . {\displaystyle \langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.} A real inner product space

4851-436: The identity. Any positive linear functionals g on A dominated by f is of the form g ( x ∗ x ) = ⟨ π ( x ) ξ , π ( x ) T g ξ ⟩ {\displaystyle g(x^{*}x)=\langle \pi (x)\xi ,\pi (x)T_{g}\,\xi \rangle } for some positive operator T g in π( A )' with 0 ≤ T ≤ 1 in

4928-539: The image of π( A ) will be a C*-algebra of operators called the C*-enveloping algebra of A . Equivalently, we can define the C*-enveloping algebra as follows: Define a real valued function on A by as f ranges over pure states of A . This is a semi-norm, which we refer to as the C* semi-norm of A . The set I of elements of A whose semi-norm is 0 forms a two sided-ideal in A closed under involution. Thus

5005-447: The introduction of Hilbert spaces. The first of these was the observation, which arose during David Hilbert and Erhard Schmidt 's study of integral equations , that two square-integrable real-valued functions f and g on an interval [ a , b ] have an inner product that has many of the familiar properties of the Euclidean dot product. In particular, the idea of an orthogonal family of functions has meaning. Schmidt exploited

5082-629: The mathematical underpinning of thermodynamics ). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis . Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions , spaces of sequences , Sobolev spaces consisting of generalized functions , and Hardy spaces of holomorphic functions . Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of

5159-549: 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. Hilbert space In mathematics , Hilbert spaces (named after David Hilbert ) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional . Hilbert spaces arise naturally and frequently in mathematics and physics , typically as function spaces . Formally,

5236-519: The operator order. This is a version of the Radon–Nikodym theorem . 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 g ⊕ π g' . This shows that π is irreducible if and only if any such π g is unitarily equivalent to π, i.e. g is a scalar multiple of f , which proves the theorem. Extremal states are usually called pure states . Note that

5313-406: The properties An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product . A vector space equipped with such an inner product is known as a (real) inner product space . Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it

5390-657: The second form (conjugation of the first element) is commonly found in the theoretical physics literature. For f and g in L , the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, L is in fact complete. The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable . The Lebesgue spaces appear in many natural settings. The spaces L ( R ) and L ([0,1]) of square-integrable functions with respect to

5467-400: The series converges in H , in the sense that the partial sums converge to an element of H . As a complete normed space, Hilbert spaces are by definition also Banach spaces . As such they are topological vector spaces , in which topological notions like the openness and closedness of subsets are well defined . Of special importance is the notion of a closed linear subspace of

5544-576: The similarity of this inner product with the usual dot product to prove an analog of the spectral decomposition for an operator of the form where K is a continuous function symmetric in x and y . The resulting eigenfunction expansion expresses the function K as a series of the form where the functions φ n are orthogonal in the sense that ⟨ φ n , φ m ⟩ = 0 for all n ≠ m . The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in

5621-522: The space L of square Lebesgue-integrable functions is a complete metric space . As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier , Friedrich Bessel and Marc-Antoine Parseval on trigonometric series easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz–Fischer theorem . Further basic results were proved in

5698-496: The space. Completeness can be characterized by the following equivalent condition: if a series of vectors ∑ k = 0 ∞ u k {\displaystyle \sum _{k=0}^{\infty }u_{k}} converges absolutely in the sense that ∑ k = 0 ∞ ‖ u k ‖ < ∞ , {\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,} then

5775-425: The turn of the 20th century, in particular spaces of sequences (including series ) and spaces of functions, can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors. In the first decade of the 20th century, parallel developments led to

5852-402: The universal representation. If Φ is the universal representation of a C*-algebra A , the closure of Φ( A ) in the weak operator topology is called the enveloping von Neumann algebra of A . It can be identified with the double dual A** . Also of significance is the relation between irreducible *-representations and extreme points of the convex set of states. A representation π on H

5929-403: Was published in 1943. Segal recognized the construction that was implicit in this work and presented it in sharpened form. 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

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