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
$k:Calk(H) \to B(H)$? 



Calkin Algebra and the embedding
(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.


