GNS construction - Knowlpedia

Let $A$ be a $C^*$ -algebra and let $\varphi$ be a positive linear functional on $A$ . The GNS construction associates a Hilbert space $H_\varphi$ , a $*$ -representation $\pi_\varphi:A\to B(H_\varphi)$ , and a vector $\xi_\varphi$ such that

\varphi(a)=\langle\pi_\varphi(a)\xi_\varphi,\xi_\varphi\rangle \quad(a\in A)

and $\pi_\varphi(A)\xi_\varphi$ is dense in $H_\varphi$ .

Cyclicity and assumptions

The triple $(\pi_\varphi,H_\varphi,\xi_\varphi)$ is a cyclic representation. The functional is bounded automatically; neither unitality of $A$ nor faithfulness of $\varphi$ is assumed.

Quotient construction

Set $N_\varphi=\{a\in A:\varphi(a^*a)=0\}$ . The formula

\langle[a],[b]\rangle_\varphi=\varphi(b^*a)

defines an inner product, linear in the first variable, on $A/N_\varphi$ . Completing gives $H_\varphi$ , and left multiplication gives $\pi_\varphi(a)[b]=[ab]$ . If $A$ is unital, then $\xi_\varphi=[1]$ . In the nonunital case, the bounded extension of $\varphi$ to the unitization, or equivalently an approximate identity, produces the canonical cyclic vector.

Example: evaluation on continuous functions

Take $A=C(X)$ for a compact Hausdorff space $X$ , choose $x_0\in X$ , and let $\varphi(f)=f(x_0)$ . This is a positive functional. With the convention that inner products are linear in the first variable,

\langle[f],[g]\rangle_\varphi =\varphi(\overline g f) =f(x_0)\,\overline{g(x_0)}.

Here $N_\varphi=\{f:f(x_0)=0\}$ , so the quotient is one-dimensional and $[f]\mapsto f(x_0)$ identifies $H_\varphi$ with $\mathbb C$ . The cyclic vector is $[1]$ , and

\begin{aligned} \pi_\varphi(h)z&=h(x_0)z,\\ \pi_\varphi(h)[1]&=[h],\\ \langle\pi_\varphi(h)[1],[1]\rangle&=h(x_0)=\varphi(h). \end{aligned}

The example makes the quotient concrete: functions indistinguishable at $x_0$ become the same vector, and the representation remembers only their value there.

Uniqueness and faithfulness

If $(\pi,H,\xi)$ is another cyclic realization of $\varphi$ , the rule $\pi_\varphi(a)\xi_\varphi\mapsto\pi(a)\xi$ extends to a unitary intertwining the two representations and carrying $\xi_\varphi$ to $\xi$ . Hence the pointed cyclic representation is unique up to a unique unitary of this form. The representation $\pi_\varphi$ need not be faithful: its kernel consists of those elements acting trivially on the quotient, a stronger condition than merely belonging to $N_\varphi$ .

States and von Neumann algebras

The zero functional gives the zero Hilbert space and the zero representation; the zero vector is cyclic there. This edge case is included by the definition, although states are usually used when a nonzero cyclic vector is desired.

When $A$ is unital, $\|\xi_\varphi\|^2=\varphi(1)=\|\varphi\|$ ; in particular a state gives a unit cyclic vector. If $A=M$ is a von Neumann algebra and $\varphi$ is normal, then $\pi_\varphi$ is a normal representation. GNS also extends to suitable densely defined weights, but the construction then starts from the left ideal of square-integrable elements rather than all of $M$ ; that extension is not part of the bounded-functional definition above.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §3.3 on the GNS construction and cyclic representations.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §3.3 on positive functionals and representations.