Gelfand-Naimark-Segal construction

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In functional analysis, given a C*-algebra A, the GNS construction establishes a correspondence between cyclic *-representations of A and certain linear functionals 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.

Contents

[edit] States and representations

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

  • π is a ring homomorphism which carries involution on A into involution on operators
  • 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.

A 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.

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

\{\pi(x)\xi:x\in A\}

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.

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.

Let π be a *-representation of a C*-algebra A on the Hilbert space H with cyclic vector ξ having norm 1. Then

x \mapsto \langle \xi, \pi(x)\xi\rangle

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.

The converse is also true. Every state on a C*-algebra is of the above type. This is the GNS construction:

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.

\rho(x)=\langle \xi, \pi(x) \xi \rangle

for every x in A.

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.

Define on A a, possibly singular, inner product

\langle x, y \rangle =\rho(x^*y).

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. The 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.

One needs to check that the action π(x)y = xy of A on itself passes through the above construction. By a C*-algebraic argument, one can show that I is a left ideal of A. Therefore 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.

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 {eλ} for A. Since positive linear functionals are bounded, the equivalence classes of the net {eλ} converges to some vector ξ in H, which is a cyclic vector for π.

It is clear that the state ρ can be recovered as a vector state on the GNS Hilbert space. This proves the theorem.

The above shows that there is a bijective correspondence between positive linear functionals and cyclic representations. Two cyclic representations πφ and πψ with corresponding positive functionals φ and ψ are unitarily equivalent if and only if φ = α ψ for some positive number α.

If ω, φ, and ψ are positive linear functionals with ω = φ + ψ, then πω is unitarily equivalent to a subrepresentation of πφ ⊕ πψ. The embedding map is given by

\pi_{\omega}(x) \xi_{\omega} \mapsto \pi_{\phi}(x) \xi_{\phi} \oplus \pi_{\psi}(x) \xi_{\psi}.

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 direc sum of corresponding irreducible GNS representations is faithful.

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.

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**.

[edit] Irreducibility

Also of significance is the relation between 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}.

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.

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 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.

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 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 the identity.

Any positive linear functionals g on A dominated by f is of the form

g(x^*x) = \langle \pi(x) \xi, \pi(x) T_g \, \xi \rangle

for some positive operator Tg in π(A)' with 0 ≤ T ≤ 1 in 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 πg ⊕ πg' . This shows that π is irreducible if and only if any such πg 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 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.

[edit] Generalizations

The Stinespring factorization theorem characterizing completely positive maps is an important generalization of the GNS construction.

[edit] References

  • William Arveson, An Invitation to C*-Algebra, Springer-Verlag, 1981
  • Jacques Dixmier, Les C*-algebres et leurs Representations, Gauthier-Villars, 1969.
    English translation: Dixmier, Jacques (1982). C*-algebras. North-Holland. ISBN 0444863915.