I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.

Let $L$ be a bounded self-adjoint linear map on a unital $C^*$ algebra $\mathcal{A}$. The following conditions are equivalent

(iii) If $y\in\mathcal{A}_+$ (set of positive semi-definite objects), $a\in\mathcal{A}$ satisfy $ya=0$, then $a^*L(y)a\geq0$.

(iv) For some full invariant set of states $S$ on $\mathcal{A}$, that $y\in\mathcal{A}_+$, $f\in S$ with $f(y)=0$ imply $fL(y)\geq0$.

My question is on condition (iii). If $L$ is a positive map, the condition holds (by definition). However if it is not, what extra condition(s) a general linear bounded map $L$ must satisfy such that condition (iii) holds.

If we replace $\mathcal{A}$ by $\mathcal{B(H)}$ for some Hilbert space $\mathcal{H}$ (finite or infinite dimensions) can we find such conditions. If some works are already in literature, please refer. Advanced thanks for any help, suggestions etc.