### Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on a Banach space $X$ we define its adjoint on an appropriate closed subspace ${X}^{\circ}$ of the norm dual ${X}^{\text{'}}$. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology $\sigma ({X}^{\circ},X)$. We give the following application: For $\Omega $ a Polish space we consider operator semigroups on the space ${\mathrm{C}}_{\mathrm{b}}\left(\Omega \right)$ of bounded, continuous functions (endowed with the compact-open topology) and on the space $\mathrm{M}\left(\Omega \right)$ of bounded Baire measures (endowed with the weak${}^{*}$-topology)....