I was reading this [paper][1] by Evans  and Hanche-Olsen. In theorem 2,  there are six equivalent statements given. I write just two of them, which I need. 

> 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.  

  [1]: http://dx.doi.org/10.1016/0022-1236(79)90054-5