Dynamical Map

$\Phi :B(H)\to B(G)$ is a Dynamical Map between trace class operators on Hilbert Spaces $H$ and $G$ if:

• $\mathrm{Tr}[\Phi (\rho )]\le 1$ for any Density Operator $\rho$.
• $\Phi$ is Completely Positive

It is unclear why we need the Dynamical Map to operate between trace class operators, nor what trace class operators are. Help me ❓