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 ( 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 π’œ be a C*-algebra and Ο• a positive linear functionalMathworldPlanetmath in π’œ.

We are going to construct a representation πϕ of π’œ and for that we need to construct a suitable Hilbert space.

Let us endow π’œ with a semi-inner product defined by ⟨x,yβŸ©Ο•:=ϕ⁒(y*⁒x). Now we define the set


It is easily seen that NΟ• is a closed left idealPlanetmathPlanetmath ( in π’œ (using the Cauchy-Schwarz inequality, which is valid in semi-inner product spaces), so that βŸ¨β‹…,β‹…βŸ©Ο• induces a well defined inner productMathworldPlanetmath on the quotient ( π’œ/NΟ•. The completion of π’œ/NΟ• is then an Hilbert space, which we will be denoted by HΟ•.

We will now define a representation of π’œ on HΟ• by left multiplicationPlanetmathPlanetmath. For every aβˆˆπ’œ let πϕ⁒(a) be the operator of left multiplication by a on π’œ/NΟ•, i.e.


Theorem 1 - The function πϕ⁒(a):π’œ/NΟ•βŸΆπ’œ/NΟ• defined above is linear and boundedPlanetmathPlanetmath (, with βˆ₯πϕ⁒(a)βˆ₯≀βˆ₯aβˆ₯.

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

Let B⁒(HΟ•) be the algebra of bounded operators on HΟ•.

Theorem 2 - The function πϕ:π’œβŸΆB⁒(HΟ•) defined by a↦πϕ⁒(a) is a C*-algebra representation of π’œ.

This representation is called the GNS representation associated to Ο•.

3 Cyclic vectors and GNS pairs

Suppose π’œ had an identity elementMathworldPlanetmath e. In this case it is easily seen that there exists a cyclic vectorMathworldPlanetmathPlanetmath ΞΎΟ•βˆˆHΟ•, i.e. a vector ΞΎΟ• such that πϕ⁒(π’œ)⁒ξϕ is dense ( in HΟ•. This cyclic vector can just be chosen as e+NΟ•.

Moreoever, this cyclic vector ΞΎΟ•:=e+NΟ• is such that ϕ⁒(a)=βŸ¨Ο€Ο•β’(a)⁒ξϕ,ΞΎΟ•βŸ©Ο• for every aβˆˆπ’œ.

Thus, in this case the representation πϕ is cyclic ( and Ο• is a vector state of π’œ. The result is still valid for general C*-algebras:

Theorem 3 - Let πϕ be the representation of π’œ defined previously. Then there exists a vector ΞΎΟ•βˆˆHΟ• such that

  • β€’

    πϕ⁒(π’œ)⁒ξϕ is dense in HΟ•, i.e. πϕ is cyclic,

  • β€’

    ϕ⁒(a)=βŸ¨Ο€Ο•β’(a)⁒ξϕ,ΞΎΟ•βŸ©Ο• for every aβˆˆπ’œ, i.e. Ο• is a vector state.

Any pair (Ο€,ΞΎ), where Ο€ is a representation of π’œ on a Hilbert space H and ξ∈H, satisfying the above conditions for Ο•:

  • β€’

    π⁒(π’œ)⁒ξ is dense in H,

  • β€’

    ϕ⁒(a)=βŸ¨Ο€β’(a)⁒ξ,ξ⟩ for every aβˆˆπ’œ

is called a GNS pair for Ο•.

Theorem 4 - All GNS pairs for Ο• 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 (, and so we have assured the existence of many (cyclic) representations. An interesting fact is that this representations associated to states are irreducible ( exactly when the state is a pure state:

Theorem 5 - Let Ο• be a state on π’œ. Then the representation πϕ is irreducible if and only if Ο• 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 π’œ.

Theorem 6 - Let π’œ be a C*-algebra. For every element a there exists an irreducible representation Ο€ of π’œ such that βˆ₯π⁒(a)βˆ₯=βˆ₯aβˆ₯.

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

Title Gelfand-Naimark-Segal construction
Canonical name GelfandNaimarkSegalConstruction
Date of creation 2013-03-22 17:47:40
Last modified on 2013-03-22 17:47:40
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Feature
Classification msc 46L30
Classification msc 46L05
Synonym GNS construction
Related topic CAlgebra
Related topic CAlgebra3
Related topic RepresentationOfAC_cG_dTopologicalAlgebra
Defines GNS pair
Defines GNS representation
Defines pure states and irreducible representations