Integrable representations of involutive algebras and Ore localization

TL;DR

This paper establishes a connection between integrable representations of involutive algebras and their Ore localizations, providing new insights into the structure and extension of such representations.

Contribution

It proves that cyclic representations can be extended integrably and that integrable representations correspond bijectively to representations of the Ore localization.

Findings

Cyclic representations admit integrable extensions.

Integrable representations correspond to Ore localization representations.

Ore localization of the algebra is an involutive algebra.

Abstract

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.