  Given any separable complex Hilbert space, any trace-class operator $B$ which
does not have purely imaginary trace, and any generator $L$ of a
norm-continuous one-parameter semigroup of completely positive maps we prove
that there exists a unique bounded operator $K$ and a unique completely
positive map $\Phi$ such that (i) $L=K(\cdot)+(\cdot)K^*+\Phi$, (ii) the
superoperator $\Phi(B^*(\cdot)B)$ is trace class and has vanishing trace, and
(iii) ${\rm tr}(B^*K)$ is a real number. Central to our proof is a modified
version of the Choi formalism which relates completely positive maps to
positive semi-definite operators. We characterize when this correspondence is
injective and surjective, respectively, which in turn explains why the proof
idea of our main result cannot extend to non-separable Hilbert spaces. In
particular, we find examples of positive semi-definite operators which have
empty pre-image under the Choi formalism as soon as the underlying Hilbert
space is infinite-dimensional.
