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.