The Gelfand–Naimark–Segal (GNS) construction establishes a correspondence between cyclic $*$-representations of $C^*$-algebras and certain linear functionals (usually called states) on those same $C^*$-algebras. The correspondence comes about from an explicit construction of the *-representation from one of the linear functionals (states).

The nPOV

The GNS construction (as outlined above) is a special case of a more general construction of Ghez, Lima and Roberts applied to $C^\ast$-categories (horizontal categorification of $C^\ast$-algebras).

Theorem

Let $\mathcal{C}$ be a $C^\ast$-category. Fix an object $A \in \operatorname{Ob}\mathcal{C}$ and let $\sigma$ be a state on the $C^\ast$-algebra $\mathcal{C}(A,A)$. Then there exists a $*$-representation

A $C*$-algebra $A$ is a $C^\ast$-category $\mathcal{A}$ with one object $\bullet$, where we make the identification $A = \mathcal{A}(\bullet,\bullet)$. In this case the theorem reduces to the classical GNS construction.