  We show that a bounded, linear map between C*-algebras is a weighted
$\ast$-homomorphism (the central compression of a $\ast$-homomorphism) if and
only if it preserves zero-products, range-orthogonality, and
domain-orthogonality. It follows that a self-adjoint, bounded, linear map is a
weighted $\ast$-homomorphism if and only if it preserves zero-products.
  As an application we show that a linear map between C*-algebras is completely
positive, order zero in the sense of Winter-Zacharias if and only if it is
positive and preserves zero-products.
