summary:For a given bi-continuous semigroup $(T(t))_{t\geq 0}$ on a Banach space $X$ we define its adjoint on an appropriate closed subspace $X^\circ $ of the norm dual $X'$. 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 ${\rm C_b}(\Omega )$ of bounded, continuous functions (endowed with the compact-open topology) and on the space ${\rm M}(\Omega )$ of bounded Baire measures (endowed with the weak$^*$-topology). We show that bi-continuous semigroups on ${\rm M}(\Omega )$ are precisely those that are adjoints of bi-continuous semigroups on ${\rm C_b}(\Omega )$. We also prove that the class of bi-continuous semigroups on ${\rm C_b}(\Omega )$ with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if $\Omega $ is not a Polish space this is not the case.
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