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单词 GelfandNaimarkSegalConstruction
释义

Gelfand-Naimark-Segal construction


1 GNS Construction

The Gelfand-Naimark-Segal construction (or GNS construction) is a fundamental idea in the of . It provides a procedure to construct and study representations of C*-algebrasMathworldPlanetmathPlanetmath (http://planetmath.org/CAlgebra) and is the first step on the proof of the Gelfand-Naimark representation theorem, which that every C*-algebra is isometrically isomorphic to a closed *-subalgebraPlanetmathPlanetmath of B(H), the algebra of bounded operatorsMathworldPlanetmathPlanetmath on a Hilbert spaceMathworldPlanetmath H.

There are generalizationsPlanetmathPlanetmath of this construction for Banach *-algebras with an approximate unitMathworldPlanetmath, and some of the results stated here are in fact valid for this kind of algebras, but we will restrict our attention to the C* case.

2 Representations associated with positive linear functionals

Let 𝒜 be a C*-algebra and ϕ a positive linear functionalMathworldPlanetmath in 𝒜.

We are going to construct a representation πϕ of 𝒜 and for that we need to construct a suitable Hilbert space.

Let us endow 𝒜 with a semi-inner product defined by x,yϕ:=ϕ(y*x). Now we define the set

Nϕ:={x𝒜:x,xϕ=0}

It is easily seen that Nϕ is a closed left idealPlanetmathPlanetmath (http://planetmath.org/Ideal) in 𝒜 (using the Cauchy-Schwarz inequality, which is valid in semi-inner product spaces), so that ,ϕ induces a well defined inner productMathworldPlanetmath on the quotient (http://planetmath.org/QuotientModule) 𝒜/Nϕ. The completion of 𝒜/Nϕ is then an Hilbert space, which we will be denoted by Hϕ.

We will now define a representation of 𝒜 on Hϕ by left multiplicationPlanetmathPlanetmath. For every a𝒜 let πϕ(a) be the operator of left multiplication by a on 𝒜/Nϕ, i.e.

π(a)(x+Nϕ):=ax+Nϕ

Theorem 1 - The function πϕ(a):𝒜/Nϕ𝒜/Nϕ defined above is linear and boundedPlanetmathPlanetmath (http://planetmath.org/BoundedOperator), with πϕ(a)a.

Being bounded, the operator πϕ(a) extends uniquely to a bounded operator on Hϕ, which we denote by the same symbol, πϕ(a).

Let B(Hϕ) be the algebra of bounded operators on Hϕ.

Theorem 2 - The function πϕ:𝒜B(Hϕ) defined by aπϕ(a) is a C*-algebra representation of 𝒜.

This representation is called the GNS representation associated to ϕ.

3 Cyclic vectors and GNS pairs

Suppose 𝒜 had an identity elementMathworldPlanetmath e. In this case it is easily seen that there exists a cyclic vectorMathworldPlanetmathPlanetmath ξϕHϕ, i.e. a vector ξϕ such that πϕ(𝒜)ξϕ is dense (http://planetmath.org/Dense) in Hϕ. This cyclic vector can just be chosen as e+Nϕ.

Moreoever, this cyclic vector ξϕ:=e+Nϕ is such that ϕ(a)=πϕ(a)ξϕ,ξϕϕ for every a𝒜.

Thus, in this case the representation πϕ is cyclic (http://planetmath.org/BanachAlgebraRepresentation) and ϕ is a vector state of 𝒜. The result is still valid for general C*-algebras:

Theorem 3 - Let πϕ be the representation of 𝒜 defined previously. Then there exists a vector ξϕHϕ such that

  • πϕ(𝒜)ξϕ is dense in Hϕ, i.e. πϕ is cyclic,

  • ϕ(a)=πϕ(a)ξϕ,ξϕϕ for every a𝒜, i.e. ϕ is a vector state.

Any pair (π,ξ), where π is a representation of 𝒜 on a Hilbert space H and ξH, satisfying the above conditions for ϕ:

  • π(𝒜)ξ is dense in H,

  • ϕ(a)=π(a)ξ,ξ for every a𝒜

is called a GNS pair for ϕ.

Theorem 4 - All GNS pairs for ϕ are (in the sense that the corresponding representations are unitarily equivalent).

4 Irreducible representations

We know that are ”plenty” of states on C*-algebra (see this entry (http://planetmath.org/PropertiesOfStates)), and so we have assured the existence of many (cyclic) representations. An interesting fact is that this representations associated to states are irreducible (http://planetmath.org/BanachAlgebraRepresentation) exactly when the state is a pure state:

Theorem 5 - Let ϕ be a state on 𝒜. Then the representation πϕ is irreducible if and only if ϕ is a pure state.

The fact that there are ”plenty” of pure states in a C*-algebra allows one to assure the existence of irreducible representations that preserve the norm of a given element in 𝒜.

Theorem 6 - Let 𝒜 be a C*-algebra. For every element a there exists an irreducible representation π of 𝒜 such that π(a)=a.

This last theorem is a fundamental step in the proof of the Gelfand-Naimark representation theorem.

TitleGelfand-Naimark-Segal construction
Canonical nameGelfandNaimarkSegalConstruction
Date of creation2013-03-22 17:47:40
Last modified on2013-03-22 17:47:40
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id10
Authorasteroid (17536)
Entry typeFeature
Classificationmsc 46L30
Classificationmsc 46L05
SynonymGNS construction
Related topicCAlgebra
Related topicCAlgebra3
Related topicRepresentationOfAC_cG_dTopologicalAlgebra
DefinesGNS pair
DefinesGNS representation
Definespure states and irreducible representations
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