Pure states and irreducible GNS representations

Statement

GNS purity criterion. Let $\varphi$ be a state on a $C^*$ -algebra $A$ , and let $(\pi_\varphi,H_\varphi,\xi_\varphi)$ be its GNS representation. Then $\varphi$ is a pure state if and only if $\pi_\varphi$ is an irreducible representation. Consequently, convex indecomposability of a state is equivalent to the absence of nontrivial closed invariant subspaces in its canonical cyclic representation. Neither side asserts that $\varphi$ or $\pi_\varphi$ is faithful.

Proof mechanism

If a nontrivial projection lies in the commutant $\pi_\varphi(A)'$ , its two orthogonal components split the vector functional into a nontrivial convex combination. Conversely, a positive functional dominated by $\varphi$ is represented by a positive contraction in that commutant; a nontrivial convex decomposition therefore produces a nonscalar commutant. Schur's lemma, $\pi_\varphi(A)'=\mathbb C I$ , completes the equivalence Murphy, Theorem 3.3.8.

Converse realization

Let $\pi:A\to\mathcal B(H)$ be irreducible and let $\xi\neq0$ . Then $\xi$ is cyclic, and after normalization its vector state is pure. The pointed GNS representation of that state is unitarily equivalent to $(\pi,H,\xi/\|\xi\|)$ . Hence every irreducible representation is obtained, up to unitary equivalence, from a pure state and a choice of nonzero vector.

Distinctions

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: Theorem 3.3.8 on pure states and irreducible GNS representations.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 3 on pure states, cyclic representations, and commutants.