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 ?

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 $A$

$\rho:A\rightarrow\mathbb{C}$

satisfying the following conditions:

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

• $\rho$ is normalized: $\rho(\mathbf{1}) = 1$.

By the Gelfand-Naimark-Segal (GNS) construction, for each such state, there is a *-representation $\pi_{\rho}$ of $A$ in a Hilbert space $\mathscr{H}_{\rho}$ and a unit vector $\Omega\in\mathscr{H}_{\rho}$ such that $$\rho(a)=\langle\pi_\rho(a)\Omega,\Omega\rangle$$ for every $a\in A$.

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