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 ?



$C^*$ algebras and states
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: 


1. $\rho$ and $\rho'$ are *pure* states (equivalently, the GNS representations $\pi\_\rho$, $\pi\_{\rho'}$ associated to $\rho$, resp. $\rho'$ are *irreducible*) or
2. $\rho$ 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](http://www.ams.org/journals/tran/1960-094-03/S0002-9947-1960-0146681-0/)), 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.


