Traceless AF embeddings and unsuspended $E$-theory

I show that quasidiagonality and AF embeddability are equivalent properties for traceless ${\mathrm{C}}^{\ast }$-algebras and are characterised in terms of the primitive ideal space. For nuclear ${\mathrm{C}}^{\ast }$-algebras the same characterisation determines when Connes and Higson’s $E$-theory can be unsuspended.