  Let $\mathcal A$ be a unital algebra equipped with an involution
$(\cdot)^\dagger$, and suppose that the multiplicative set $\mathcal S\subseteq
\mathcal A$ generated by the elements of the form $1 + a^\dagger a$ satisfies
the Ore condition. We prove that: (i) Cyclic representations of $\mathcal A$
admit an integrable extension (acting on a possibly larger Hilbert space), and
(ii) Integrable representations of $\mathcal A$ are in bijection with
representations of the Ore localization $\mathcal A\mathcal S^{-1}$ (which we
prove to be an involutive algebra). This second result is a limited converse to
a theorem by Inoue asserting that representations of symmetric involutive
algebras are integrable.
