Gelfand-Naimark-Segal construction

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, although the designation of construction usually just refers to the second part of that theorem.

States and representations

A *-representation of a C*-algebra on a Hilbert space H is a *-morphism π from A into the algebra of bounded operators on H which 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, non-degeneracy means exactly π is unit-preserving.

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.

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.

A representation π of a C*-algebra A on a Hilbert space H has cyclic vector ξ iff the set of vectors

<math>\{\pi(x)\xi:x\in A\}<math>

is norm dense in H. 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.

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

<math> x \mapsto \langle \xi, \pi(x)\xi\rangle <math>

is a state of A. Given *-representations π, π' each with unit norm cyclic vectors ξ, ξ' and having the same associated states, then π, π' are unitarily equivalent representations; moreover, the unitary operator U that implements the unitary equivalence can be chosen to map ξ to ξ'.

Conversely, given a state ρ there is a *-representation π of A with distinguished cyclic vector ξ such that its associated state is ρ

<math>\rho(x)=\langle \xi, \pi(x) \xi \rangle<math>

for every x in A.

The construction proceeds as follows: Assume A has a unit element. A can be equipped with a singular inner product

<math> \langle x, y \rangle =\rho(x^*y)<math>

Here singular means that the sesquilinear form fails to satisfy the non-degeneracy property of inner product. However, we take the quotient space of the vector subspace A by the vector subspace I consisting of elements x of A satisfying ρ(x* x)=0. The vector space I is actually a left ideal of A, so the elements of A act on A/I as operators on the left. H is then taken to be the Cauchy completion of A/I, equipped with the quotient norm. The cyclic vector ξ is the image of 1 in A/I. In case A does not have a unit element, consider the C*-algebra A₁ obtained from A by adjoining a multiplicative identity. Any state f on A extends uniquely to a state f₁ on A₁. Apply the previous construction to f₁.

This construction is at the heart of the proof of the Gelfand-Naimark theorem characterizing C*-algebras as algebras of operators.

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 iff 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. 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 that given a self-adjoint operator T on H which commutes with all the operators π(x), and is such that 0 ≤ T ≤ 1 in the operator order,

<math> f_T(x) = \langle T \xi, \pi(x) T\xi \rangle <math>

is a positive linear functional on A (not in general a state) dominated by f. This map is easily shown to be a bijection. Now the representation π is irreducible iff the only operators which commute with all the π(x) are scalar multiples of the identity. Thus a necessary and sufficient condition π be irreducible is that the set of states dominated by f consist only of scalar multiples of f. This condition on f can be shown to be equivalent to f being an extreme point in the set of positive linear functionals of norm ≤ 1.

Extremal states are usually called pure states. Note that a state is a pure state iff 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.

References

• William Arveson, An Invitation to C*-Algebra, Springer-Verlag, 1981
• Jacques Dixmier, Les C*-algebres et leurs Representations, Gauthier-Villars, 1969