  Let $G$ be a connected Lie group and $\hat G$ its unitary dual. We are
interested in the part $\Lambda\subset\hat G$ which corresponds to the unitary
highest weight representations of $G$. Then there are several topologies on
$\Lambda$: The euclidean topology $T_E$ which comes from the identification of
$\Lambda$ with the set of highest weights, the induced topology $T_I$ induced
from the Fell topology on $\hat G$ and finally a natural topology $T_S$ which
comes from the hull kernel topology of certain CCR C^*-algebras which are
related to the holomorphic extemsion of unitary highest weight representations
to complex Olshanskii semigroups $S$. One of the main results in this paper is
the inclusion chain $T_S\subset T_I\subset T_E$. Further we exhibit very large
interesting subspaces of $\Lambda$ where these topologies coincide. Finally we
show that the Borel structures on $\Lambda$ induced from the three different
topologies coincide.
