Trivial Endomorphisms of the Calkin Algebra

Abstract

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