  We extend the theory of holomorphic induction of unitary representations of a
possibly infinite-dimensional Lie group $G$ beyond the setting where the
representation being induced is required to be norm-continuous. We allow the
group $G$ to be a connected regular BCH(Baker-Campbell-Hausdorff) Fr\'echet-Lie
group. Given a smooth $\mathbb{R}$-action $\alpha$ on $G$, we proceed to show
that the corresponding class of so-called positive energy representations is
intimately related with holomorphic induction. Assuming that $G$ is regular, we
in particular show that if $\rho$ is a unitary ground-state representation of
$G \rtimes_\alpha \mathbb{R}$ for which the energy-zero subspace
$\mathcal{H}_\rho(0)$ admits a dense set of $G$-analytic vectors, then
$\rho\big|_G$ is holomorphically induced from the representation of the
connected subgroup $H := (G^\alpha)_0$ of $\alpha$-fixed points on
$\mathcal{H}_\rho(0)$. As a consequence, we obtain an isomorphism
$\mathcal{B}(\mathcal{H}_\rho)^G \cong \mathcal{B}(\mathcal{H}_\rho(0))^H$
between the corresponding commutants. We also find that any two such
ground-state representations are necessarily unitary equivalent if their
energy-zero subspaces are unitarily equivalent as $H$-representations. These
results were previously only available under the assumption of norm-continuity
of the $H$-representation on $\mathcal{H}_\rho(0)$.
