  In this study, we refine the compactification presented by Witz \cite{Witz}
for general semigroups to the case of bounded $C_0$-semigroups, involving
adjoint theory for this class of operators. This approach considerably reduces
the operator space in which the compactification is performed. Additionally,
this approach leads to a decomposition of $X^{\sun}$ and to an extension of
ergodic results to dual semigroups.
