Trivial endomorphisms of the Calkin algebra

Abstract

We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra \({\cal Q}(H)\) is unitarily equivalent to an endomorphism of \({\cal Q}(H)\) which is liftable to a unital endomorphism of \({\cal B}(H)\). We use this result to classify all unital endomorphisms of \({\cal Q}(H)\) 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 C*-algebras that embed into \({\cal Q}(H)\) is not closed under tensor product nor countable inductive limit.