I have a question arising from von Neumann's $C^*$ algebra formulation of quantum mechanics. In it, a state is a $\mathbb{C}$-linear functional on a $C^*$ algebra $A$,

$ρ:A \rightarrow \mathbb{C}$, such that,

it is positive, i.e., for every $a\in A$, we have $ \rho(a^{*} a) \geq 0 \in \mathbb{R}$;

it is normalized: $\rho(1) = 1$.

By Gelfand-Naimark-Segal (GNS) construction, for each such state, there is a Hilbert space $*$-representation $\nu$ of $A$ and a unit vector $u$ in the Hilbert space such that, for every $x\in A$ $$\rho(x)=\langle \nu[x](u)\,|\,u\rangle\ .$$

Question: what conditions do two states $\rho, \rho'$ have to satisfy such that the associated representations be equivalent ?