Positive energy representations and continuity of projective representations for general topological groups

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 $G^\sharp$.