Integrable representations of involutive algebras and Ore localization · arXivDesk
1012.4435Dec 20, 2010Final 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
Integrable representations of involutive algebras and Ore localization
Let A be a unital algebra equipped with an involution (⋅)†, and suppose that the multiplicative set S⊆A generated by the elements of the form 1+a†a
Nearby in the stack
satisfies the Ore condition. We prove that: (i) Cyclic representations of
A
admit an integrable extension (acting on a possibly larger Hilbert space), and (ii) Integrable representations of
A
are in bijection with representations of the Ore localization
AS−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.