This is a more specific version of my last question. Let $H$ be an infinite dimensional Hilbert space. Is the Calkin algebra $Q(H)=B(H)/K(H)$ linearly homeomorphic to a dual space in general? If not, is it for any particular $H$? I suspect that for separable $H,$ it is not. This intuition comes from the fact that $\ell^\infty/ c_0$ (which can be isometrically embedded as the diagonal operators) is not isomorphic to a dual space.
As noted in my last question, such a homeomorphism is never *-isomorphic, but for my purposes, this does not matter.
As I noted in this answer, if we have a Banach space $F$ s.t. $F^*$ is linearly homeomorphic to $Q(H),$ then $F$ corresponds to a minimal closed, point separating subspace of $Q(H)^*.$ This then corresponds to a minimal locally convex Hausdorff topology on $Q(H)$ (namely $\sigma(Q(H),F)$) whose continuous dual is Banach in the canonical way.
