Integrable representations of involutive algebras and Ore localization

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.