Let $H$ be a separable, infinite dimensional Hilbert Space and $Calk(H):=B(H)/K(H)$ denotes the Calkin algebra. There is obvious surjection $\pi: B(H) \to Calk(H)$ but I'm interested in somehow opposite question: is it possible to construct embedding $j_1:Calk(H) \to B(H)$? The same question for embedding $j_2:B(H) \to Calk(H)$ and finally is there epimorphism $k:Calk(H) \to B(H)$?
(1) No. The Calkin algebra contains an uncountable family of mutually orthogonal nonzero projections.
(2) Yes. Embed $B(H)$ into $B(H \otimes H)$ by the map $A \mapsto A \otimes I$, then pass to $Q(H\otimes H) \cong Q(H)$.
(3) No. The Calkin algebra is simple.
(1) One cannot embed $Q(H)$ into $B(H)$ as a Banach space either. Indeed, $Q(H)$ contains $\ell_\infty/c_0$ (as a diagonal masa). J. Bourgain gave a very clever proof of the fact that $\ell_\infty/c_0$ has no strictly convex renorming:
J. Bourgain, $\ell_\infty/c_0$ has no equivalent strictly convex norm, Proc. Amer. Math. Soc. 78 (1980), 225-226.
On the other hand, $B(H)$ is a dual space of a separable Banach space (the space of nuclear operators on $H$), so it does have a strictly convex renorming. The property of having a strictly convex renorming passes to subspaces.