  We say a completely positive contractive map between two C*-algebras has
order zero, if it sends orthogonal elements to orthogonal elements. We prove a
structure theorem for such maps. As a consequence, order zero maps are in
one-to-one correspondence with *-homomorphisms from the cone over the domain
into the target algebra. Moreover, we conclude that tensor products of order
zero maps are again order zero, that the composition of an order zero map with
a tracial functional is again a tracial functional, and that order zero maps
respect the Cuntz relation, hence induce ordered semigroup morphisms between
Cuntz semigroups.
