$C^*$ algebras and states 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 ?
 A: If you want a criterion which is not tautological, that is, beyond the very definition of equivalence of *-representations, there are (at least) two situations where there is a criterion for equivalence of GNS representations, namely: 


*

*$\rho$ and $\rho'$ are pure states (equivalently, the GNS representations $\pi_\rho$, $\pi_{\rho'}$ associated to $\rho$, resp. $\rho'$ are irreducible) or 

*$\rho$ and $\rho'$ are type III factor states (i.e. the von Neumann algebras $\pi_\rho(A)''$, $\pi_{\rho'}(A)''$ are type III factors) and the GNS Hilbert spaces $\mathscr{H}_\rho$, $\mathscr{H}_{\rho'}$ are both separable. 


In both cases, $\pi_\rho$ and $\pi_{\rho'}$ are equivalent iff they are quasi-equivalent, i.e. neither representation has a nonzero subrepresentation disjoint from the other. Generally, $\pi_\rho$ and $\pi_{\rho'}$ are quasi-equivalent iff there are trace-class positive linear operators $T_\rho\in\mathfrak{B}(\mathscr{H}_{\rho'})$, $T_{\rho'}\in\mathfrak{B}(\mathscr{H}_\rho)$ with unit trace (i.e. density matrices) such that $\rho(a)=\mathrm{Tr}(T_\rho\pi_{\rho'}(a))$ and $\rho'(a)=\mathrm{Tr}(T_{\rho'}\pi_\rho(a))$ for all $a\in A$. By Fell's equivalence theorem (Theorem 1.2 of J.M.G. Fell, The Dual Spaces of C*-Algebras, Trans. Amer. Math. Soc. 94 (1960) 365-403), quasi-equivalence of $\pi_\rho$ and $\pi_{\rho'}$ also amounts to requiring: 


*

*$\ker\pi_\rho=\ker\pi_{\rho'}=J$, and 

*The identity map of $A/J$ extends to a *-isomorphism from $\pi_\rho(A)''$ onto $\pi_{\rho'}(A)''$. 


Therefore, these two conditions together may be thought of as an equivalence criterion in situations 1.) and 2.) above.
A: Your title should not include normal, since that applies only to states on vN algebras (and possibly AW$*$ algebras). 
Decades ago, George Elliott proved that if $A$ were a simple (and unital---all C$*$-algebras are unital here) AF C$*$-algebra, then any two extremal (aka pure) states were conjugate (via an automorphism, I think it turned out to be approximately inner). This was subsequently generalized to a much larger class, including non-simple, where the criterion applied to two extremal traces \st the kernels of the representations were equal. In this case, conjugacy is enough for equivalence (although it is not in general, consider the transposition on $\mathbf C \times \mathbf C$). 
For impure states, there is not going to be much of a result, unless you know their integral decompositions.
