Integrable Representations of Involutive Algebras and Ore Localization

Abstract

Let ${\mathcal A}$ be a unital algebra equipped with an involution (·)†, and suppose that the multiplicative set ${\mathcal S}\subseteq {\mathcal A}$ generated by the elements of the form 1 + a † a contains only regular elements and satisfies the Ore condition. We prove that ultracyclic representations of ${\mathcal A}$ admit an integrable extension, and that integrable representations of ${\mathcal A}$ are in bijection with representations of the Ore localization ${\mathcal A}\mathcal S^{-1}$ (which is an involutive algebra). This second result can be understood as a restricted converse to a theorem by Inoue asserting that representations of symmetric involutive algebras are integrable.