Positive Projections on Matrix Subspaces

Question

Let $\mathcal{M}_d$ be the space of complex valued $d\times d$ matrices, and $\mathcal{S}\subset \mathcal{M}_d$ a subspace. Let $P_{\mathcal{S}}$ be a projection onto $\mathcal{S}$. My questions is about what properties must $\mathcal{S}$ fulfill to guarantee that there exists a $P_{\mathcal{S}}$ with the property of being positive preserving. Example: if $\mathcal{S}={\rm span}(A)$ where $A$ a positive semidefinite matrix, then by using the Hilbert-Schmidt inner product, one constructs a positive preserving $P_{\mathcal{S}}$ (completely positive actually).