GNS uniqueness theorem - Imported Knowlpedia Content

Statement

Let $\varphi$ be a positive linear functional on a $C^*$ -algebra $A$ . Suppose $(\pi_\varphi,H_\varphi,\xi_\varphi)$ is the cyclic triple produced by the GNS construction, and $(\rho,K,\eta)$ is any cyclic representation satisfying

\varphi(a)=\langle\rho(a)\eta,\eta\rangle \qquad(a\in A).

Then there is a unique unitary $U:H_\varphi\to K$ such that

U\xi_\varphi=\eta \quad\text{and}\quad U\pi_\varphi(a)=\rho(a)U \qquad(a\in A).

Thus the pointed cyclic representation implementing $\varphi$ is unique up to canonical unitary equivalence. The uniqueness is pointed: both the representation and its distinguished cyclic vector are part of the data.

Construction of the unitary

On the dense cyclic subspace, define

U_0\bigl(\pi_\varphi(a)\xi_\varphi\bigr)=\rho(a)\eta.

The implementing identities show that both sides have the same inner products:

\langle\pi_\varphi(a)\xi_\varphi,\pi_\varphi(b)\xi_\varphi\rangle =\varphi(b^*a) =\langle\rho(a)\eta,\rho(b)\eta\rangle.

Hence $U_0$ is well defined and isometric. Cyclicity makes its range dense, so it extends to the asserted unitary. Density also proves uniqueness Murphy, Theorem 3.3.3.

Why the pointing matters

The theorem is stronger than unpointed unitary equivalence: the implementing unitary must carry the distinguished cyclic vector to the other distinguished vector. An unpointed cyclic representation may have many cyclic vectors that induce different positive functionals. Likewise, dropping cyclicity permits extra orthogonal summands and destroys uniqueness.

References

Gerard J. Murphy, C-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §3.3, especially Theorem 3.3.3 on existence and uniqueness of the cyclic representation.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 3 on positive functionals and cyclic representations.