Integrable representations of involutive algebras and Ore localization

Abstract

Let $\mathcal{A}$ be a unital algebra equipped with an involution $(\cdot {)}^{†}$, and suppose that the multiplicative set $\mathcal{S}\subseteq \mathcal{A}$ generated by the elements of the form $1+a^{†}a$ 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.

Final version, to be published in Algebras and Representation Theory. Section 2 shortened, proof of Corollary 3.11 (now 3.12) corrected, and other minor changes

Cited by in corpus (1)

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