  Let $G$ and $T$ be topological groups, $\alpha : T \to \Aut(G)$ a
homomorphism defining a continuous action of $T$ on $G$ and $G^\sharp := G
\rtimes_\alpha T$ the corresponding semidirect product group. In this paper we
address several issues concerning irreducible continuous unitary
representations $(\pi^\sharp, \cH)$ of $G^\sharp$ whose restriction to $G$
remains irreducible. First we prove that, for $T = \R$, this is the case for
any irreducible positive energy representation of $G^\sharp$, i.e., for which
the one-parameter group $U_t := \pi^\sharp(\1,t)$ has non-negative spectrum.
The passage from irreducible unitary representations of $G$ to representations
of $G^\sharp$ requires that certain projective unitary representations are
continuous. To facilitate this verification, we derive various effective
criteria for the continuity of projective unitary representations. Based on
results on Borchers for $W^*$-dynamical systems, we also derive a
characterization of the continuous positive definite functions on $G$ that
extend to a $G^\sharp$.
