Trivial Endomorphisms of the Calkin Algebra

We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra $\mathcal{Q}(H)$ is unitarily equivalent to an endomorphism of $\mathcal{Q}(H)$ which is liftable to a unital endomorphism of $\mathcal{B}(H)$. We use this result to classify all unital endomorphisms of $\mathcal{Q}(H)$ up to unitary equivalence by the Fredholm index of the image of the unilateral shift. As a further application, we show that it is consistent with ZFC that the class of $\mathrm{C}^\ast$-algebras that embed into $\mathcal{Q}(H)$ is not closed under tensor product nor countable inductive limit.