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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Antoine, J.P., Inoue, A., Trapani, C.: Partial *-algebras and their Operator Realizations. Kluwer, Boston (2002)
Bagarello, F.: Algebras of unbounded operators and physical applications: a survey. Rev. Math. Phys. 19, 231–271 (2007)
Bhatt, S.J., Inoue, A., Ogi, H.: Unbounded C*-seminorms and unbounded C*-spectral algebras. J. Oper. Theory 45(1), 53–80 (2001)
Borchers, H.J., Yngvason, J.: Partially conmmutative moment problems. Math. Nachr. 145, 111–117 (1990)
Conway, J.B.: A Course in Functional Analysis. Springer, Berlin Heidelberg New York (1994)
Dubin, D.A., Hennings, M.A.: Quantum Mechanics, Algebras and Distributions. Pitman Research Notes in Mathematics, vol. 238. Longman Scientific and Technical (1990)
Fragoulopoulou, M.: Topological Algebras with Involution. North-Holland Mathematical Studies, vol. 200. Elsevier, Amsterdam (2005)
Goodearl, K.R., Warfield, R.B.: An Introduction to Noncommutative Noetherian Rings, 2nd edn. Cambridge University Press, Cambridge (2004)
Inoue, A.: On a class of unbounded operator algebras. Pac. J. Math. 65(1), 77–95 (1976)
Inoue, A.: Unbounded representations of symmetric *-algebras. J. Math. Soc. Jpn. 29(2), 219–232 (1977)
Inoue, A.: Well-behaved *-representations of *-algebras. Acta Univ. Ouluensis., A Sci. Rerum Nat. 408, 107–117 (2004)
Mallios, A., Haralampidou, M.: Topological Algebras and Applications. American Mathematical Society (2005)
Powers, R.T.: Self-adjoint algebras of unbounded operators. Commun. Math. Phys. 21, 85–124 (1971)
Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Birkhäuser, Cambridge (1990)
Schmüdgen, K.: On well-behaved unbounded representations of *-algebras. J. Oper. Theory 48(3), 487–502 (2002)
Segal, I.: A non-commutative extension of abstract integration. Ann. Math. 57(3), 401–457 (1953)
Trapani, C.: Unbounded C*-seminorms, biweights, and *-representations of *-algebras: a review. Int. J. Math. Math. Sci. 2006, 1–34 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Fondecyt Postdoctoral Grant N°3110045.
Rights and permissions
About this article
Cite this article
Vargas Le-Bert, R. Integrable Representations of Involutive Algebras and Ore Localization. Algebr Represent Theor 15, 1147–1161 (2012). https://doi.org/10.1007/s10468-011-9283-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-011-9283-5