On Compactifications of bounded $C_0-$semigroups

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.