The GNS Construction

Introduction

The GNS construction, after Gelfand, Naimark and Segal, turns a state into a representation. From a positive normalized linear functional $\omega$ on a unital $\ast$-algebra $A$ one forms the sesquilinear form $\langle x,y\rangle_{\omega} = \omega(y^{\dagger}x)$, which is positive semidefinite by the positivity of the state; one divides by its radical $N_{\omega}$, completes the quotient to a Hilbert space $H_{\omega}$, and lets $A$ act by left multiplication. The class of the unit is a cyclic vector $\xi_{\omega}$, the representation $\pi_{\omega}$ is a $\ast$-representation, and the state is recovered as the diagonal matrix element $\omega(x) = \langle\pi_{\omega}(x)\xi_{\omega},\xi_{\omega}\rangle$. Every $\ast$-representation with a cyclic vector arises this way, up to unitary equivalence, so the construction is a dictionary between states and cyclic representations.

The construction is short, and its consequences are not: applied to the algebra itself it produces the universal representation and hence the Gelfand–Naimark theorem; applied to a single state it produces the cyclic representation of that state; and the irreducibility criterion says that the state is pure exactly when its representation is irreducible, which turns a statement about functionals into a statement about lattices of invariant subspaces. This article fixes the construction, the properties of the representation, the uniqueness up to unitary equivalence and the irreducibility criterion.

The positive functionals and their positivity are Hilbert Algebras; the indefinite version, where positivity is dropped and the space becomes a Krein space, is The Indefinite GNS Construction; the completion of a Hilbert algebra is The Completion of a Hilbert Algebra; the operator algebras generated by the representation are The Left and the Right Regular Representation and Von Neumann Algebras and the Hilbert Algebra Completeness. Those are cited. The algebra is $A$, the state is $\omega$, and the constructed objects are $H_{\omega}$, $\pi_{\omega}$, $\xi_{\omega}$.

States and the Form

Definition. A state on a unital $\ast$-algebra $A$ is a linear functional $\omega : A\to\mathbb{C}$ with

$$ \omega(x^{\dagger}x)\geq0 \text{ for every } x, \qquad \omega(1) = 1 . $$

The form of the state is $\langle x,y\rangle_{\omega} = \omega(y^{\dagger}x)$.

Proposition (positivity and the Cauchy–Schwarz inequality). The form of a state is positive semidefinite, Hermitian, and satisfies

$$ |\langle x,y\rangle_{\omega}|^{2}\leq \langle x,x\rangle_{\omega}\langle y,y\rangle_{\omega}, \qquad \langle x,y\rangle_{\omega} = \overline{\langle y,x\rangle_{\omega}} . $$

Proof. The classical Cauchy–Schwarz argument for a positive semidefinite sesquilinear form: positivity of $\langle x + \lambda y, x+\lambda y\rangle_{\omega}$ for all complex $\lambda$ gives the inequality; Hermitian symmetry is $\omega(y^{\dagger}x) = \overline{\omega(x^{\dagger}y)}$, using $\omega(u^{\dagger}) = \overline{\omega(u)}$ for a state.

Definition. The radical of the state is $N_{\omega} = \{x : \langle x,x\rangle_{\omega} = 0\}$, equivalently $\{x : \omega(y^{\dagger}x) = 0 \text{ for every } y\}$ by the inequality.

Proposition (the radical is a left ideal and $\dagger$-closed). Written as $N_{\omega} = \{x : \omega(y^{\dagger}x) = 0 \text{ for every } y\}$, the radical is a left ideal and is closed under the involution.

Proof. If $x\in N_{\omega}$ and $z\in A$ then for every $y$ one has $\omega(y^{\dagger}zx) = \omega((z^{\dagger}y)^{\dagger}x) = 0$, so $zx\in N_{\omega}$. If $x\in N_{\omega}$ then for every $y$ one has $\omega(y^{\dagger}x^{\dagger}) = \overline{\omega(xy)} = \overline{\omega((y^{\dagger})^{\dagger}x)} = 0$, so $x^{\dagger}\in N_{\omega}$.

The Construction

Theorem (GNS). Let $\omega$ be a state on $A$. Then:

$A/N_{\omega}$ is a pre-Hilbert space for the induced form, whose completion $H_{\omega}$ is a Hilbert space;
the left multiplication descends to a $\ast$-representation $\pi_{\omega} : A\to B(H_{\omega})$, $\pi_{\omega}(x)(y+N_{\omega}) = xy + N_{\omega}$, which is non-degenerate and normal;
the class $\xi_{\omega} = 1 + N_{\omega}$ is cyclic, $\overline{\pi_{\omega}(A)\xi_{\omega}} = H_{\omega}$, and

$$ \omega(x) = \langle\pi_{\omega}(x)\xi_{\omega},\xi_{\omega}\rangle, \qquad \langle\pi_{\omega}(x)\xi_{\omega},\pi_{\omega}(y)\xi_{\omega}\rangle = \omega(y^{\dagger}x). $$

Proof. The form descends because the radical is exactly the isotropic part; completion gives $H_{\omega}$; the left multiplication is well defined because $N_{\omega}$ is a left ideal, is multiplicative by associativity, and satisfies $\pi_{\omega}(x^{\dagger}) = \pi_{\omega}(x)^{*}$ by the adjoint axiom and the reality of the form; the density of the image of the quotient is the construction; the two identities are the definitions.

Definition. The triple $(H_{\omega},\pi_{\omega},\xi_{\omega})$ is the GNS representation of the state $\omega$.

Properties and Uniqueness

Proposition (positivity of the representation). $\pi_{\omega}$ is a $\ast$-representation, it is non-degenerate, and it is bounded with $\|\pi_{\omega}(x)\|\leq$ the seminorm bound; the vector $\xi_{\omega}$ has norm one.

Proof. The representation identities are above; $\|\xi_{\omega}\|^{2} = \omega(1) = 1$.

Theorem (uniqueness up to unitary equivalence). If $(H,\pi,\xi)$ is a cyclic $\ast$-representation of $A$ with

$$ \omega(x) = \langle\pi(x)\xi,\xi\rangle $$

then there is a unitary $U : H_{\omega}\to H$ with $U\xi_{\omega} = \xi$ and $U\pi_{\omega}(x) = \pi(x)U$ for every $x$. So the state determines its cyclic representation, and the map from cyclic representations to states is a bijection on unitary equivalence classes.

Proof. The linear map $U(\pi_{\omega}(x)\xi_{\omega}) = \pi(x)\xi$ is well defined because the two forms agree, is isometric by the identity $\langle\pi(x)\xi,\pi(y)\xi\rangle = \omega(y^{\dagger}x)$, and extends by cyclicity; it is surjective by cyclicity of $\xi$ and intertwines the representations by multiplicativity.

Proposition (the canonical vector is separating exactly when the representation is faithful). $\xi_{\omega}$ is separating for $\pi_{\omega}(A)$ if and only if $\omega(y^{\dagger}x) = 0$ for all $y$ forces $x = 0$ in $A$.

Proof. $\pi_{\omega}(x)\xi_{\omega} = 0$ means $x\in N_{\omega}$; separatingness is the triviality of the radical.

The Irreducibility Criterion

Definition. The state $\omega$ is pure when it is an extreme point of the convex set of states; the representation $\pi_{\omega}$ is irreducible when no proper closed subspace is invariant.

Theorem (irreducibility criterion). $\omega$ is pure if and only if $\pi_{\omega}$ is irreducible.

Proof. If $\pi_{\omega}$ is reducible, a projection $P$ commuting with $\pi_{\omega}(A)$ and not $0$ or $1$ decomposes $\omega$ as a convex combination $\omega = \lambda\omega_1 + (1-\lambda)\omega_2$ with $\lambda = \|\xi_1\|^{2}$, and $\omega$ is not an extreme point. Conversely, a decomposition $\omega = \lambda\omega_1 + (1-\lambda)\omega_2$ gives subrepresentations whose cyclic vectors lie in orthogonal invariant subspaces, so $\pi_{\omega}$ has a reducing subspace.

Remark (why the criterion matters). The criterion converts the geometric question — is a state an extreme point — into the operator question — does the representation commute with a nontrivial projection. It is the entrance to the theory of factor representations, to the lattice of projections of the commutant and to the type classification, and it is the reason states and representations can be studied interchangeably. Applied to the algebra of all states it is the source of the Gelfand–Naimark theorem.

Worked Cases

The Abelian Case

For an abelian $\ast$-algebra a pure state is a character, $\omega(xy) = \omega(x)\omega(y)$, the GNS space is one-dimensional and $\pi_{\omega}$ is the character itself; this is the Gelfand theory of a commutative $\ast$-algebra in its simplest form.

Matrices

For $A = M_n(\mathbb{C})$ a state is a density matrix $\omega(x) = \mathrm{tr}(\rho x)$ with $\rho\geq0$, $\mathrm{tr}\rho = 1$; the GNS space is $\mathbb{C}^{n}$ with the standard form when $\rho$ is faithful, the representation is the defining one, and the state is pure exactly when $\rho$ is a rank-one projection.

The Trivial State

On any $A$ the state $\omega(x) = \lambda(x)$ for a character $\lambda$ produces the one-dimensional representation $\pi_{\omega}(x) = \lambda(x)$; the radical is the kernel of $\lambda$, and the criterion is the statement that a character is a pure state.

Summary

The GNS construction attaches to a state $\omega$ on a unital $\ast$-algebra the form $\langle x,y\rangle_{\omega} = \omega(y^{\dagger}x)$, positive semidefinite with radical $N_{\omega}$ — a left ideal and $\dagger$-closed set — and completes the quotient $A/N_{\omega}$ to a Hilbert space $H_{\omega}$ on which the left multiplication is a $\ast$-representation $\pi_{\omega}$ with cyclic vector $\xi_{\omega} = 1 + N_{\omega}$; the state is recovered by $\omega(x) = \langle\pi_{\omega}(x)\xi_{\omega},\xi_{\omega}\rangle$ and its values on products by $\langle\pi_{\omega}(x)\xi_{\omega},\pi_{\omega}(y)\xi_{\omega}\rangle = \omega(y^{\dagger}x)$. Every cyclic $\ast$-representation is unitarily equivalent to the GNS representation of its vector state, so states and cyclic representations correspond, and the vector is separating exactly when the representation is faithful. The irreducibility criterion states that $\omega$ is pure if and only if $\pi_{\omega}$ is irreducible, turning a statement about the convex set of states into a statement about invariant subspaces and opening the way to the structure theory of factors. The positivity is Hilbert Algebras, the indefinite counterpart is The Indefinite GNS Construction, the completion of a Hilbert algebra is The Completion of a Hilbert Algebra, and the generated algebras are The Left and the Right Regular Representation and Von Neumann Algebras and the Hilbert Algebra Completeness.

Summary of Notation

Symbol	Meaning
$\omega$, $\omega(x^{\dagger}x)\geq0$, $\omega(1)=1$	State
$\langle x,y\rangle_{\omega} = \omega(y^{\dagger}x)$	The form of the state
$N_{\omega}$	Radical, a left ideal, $\dagger$-closed
$H_{\omega} = \overline{A/N_{\omega}}$	GNS Hilbert space
$\pi_{\omega}(x)(y+N_{\omega}) = xy+N_{\omega}$	The GNS representation, a $\ast$-representation
$\xi_{\omega} = 1+N_{\omega}$	Cyclic vector of norm one
$\omega(x) = \langle\pi_{\omega}(x)\xi_{\omega},\xi_{\omega}\rangle$	Recovery of the state
$\omega$ pure $\iff$ $\pi_{\omega}$ irreducible	The irreducibility criterion