  We will give an abstract characterization of an arbitrary self-adjoint
weak$^*$-closed subspace of $\mathcal{L}(H)$ (equipped with the induced matrix
norm, the induced matrix cone and the induced weak$^*$-topology). In order to
do this, we obtain a matrix analogues of a result of Bonsall for $^*$-operator
spaces equipped with closed matrix cones. On our way, we observe that for a
$^*$-vector $X$ equipped with a matrix cone (in particular, when $X$ is an
operator system or the dual space of an operator system), a linear map
$\phi:X\to M_n$ is completely positive if and only if linear functional
$[x_{i,j}]_{i,j}\mapsto \sum_{i,j=1}^n \phi(x_{i,j})_{i,j}$ on $M_n(X)$ is
positive.
