Abstract
In this paper, C*-extensions of the Toeplitz algebras by isometric operators are investigated. It is shown that when the action of the Toeplitz algebra is irreducible, all such extensions generate the same algebra, i.e., there are no non-trivial extensions of the Toeplitz algebra. Examples of non-trivial extensions of the Toeplitz algebra are given in the case when its representation is reducible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. A. Coburn, “The C*-algebra generated by an isometry,” Bull. Amer. Math. Soc. 73, 722–726 (1967).
R. G. Douglas, “On the C*-algebra of a one-parameter semigroup of isometries,” Acta Math. 128(1), 143–152 (1972).
G. J. Murphy, “Ordered groups and Toeplitz algebras,” J. Operator Theory 18(2), 303–326 (1987).
S. Y. Jang, “Uniqueness property of C*-algebras like the Toeplitz algebras,” Trends Math. 6(2), 29–32 (2003).
I. Raeburn and S. T. Vittadello, “The isometric representation theory of a perforated semigroup,” J. Operator Theory 62(2), 357–370 (2009).
S. A. Grigoryan and V. H. Tepoyan, “On isometric representations of the perforated semigroups,” Lobachevskii J. Math. 34(1), 85–88 (2013).
V. H. Tepoyan, “On isometric representations of the semigroup ℤ+\{1},” J. Contemp. Math. Anal. 48(2), 51–57 (2013).
S. Y. Jang, “Reduced crossed products by semigroups of automorphisms,” Korean Math. Soc. 36, 97–107 (1999).
S. A. Grigoryan and A. F. Salakhutdinov, “C*-algebras generated by semigroups,” Russian Math. (Iz. VUZ) 53(10), 61–63 (2009).
S. Y. Jang, “Generalized Toeplitz algebras of a certain non-amenable semigroup,” Bull. Korean Math. Soc. 43(2), 333–341 (2006).
M. A. Aukhadiev and V. H. Tepoyan, “Isometric representations of totally ordered semigroups,” Lobachevskii J. Math. 33(3), 39–243 (2012).
S. A. Grigoryan and A. F. Salakhutdinov, “C*-algebras generated by cancellative semigroups,” Sib. Math. J. 51(1), 12–19 (2010).
A. H. Clifford and G. B. Preston, The Algebraic Theory of Semogroups, Vol. 1 (AMS, Providence, 1964; Mir, Moscow, 1972).
J.B. Garnett, Bounded Analytic Functions (Academic, New York, 1981; Mir, Moscow, 1984).
V. A. Arzumanyan, “*-representations of inverse semigroups, Izv. Akad. Nauk Arm. SSR, 13(2), 107–113 (1978).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © T.A. Grigoryan, E.V. Lipacheva, V.H. Tepoyan, 2012, published in Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2012, Vol. 154, No. 4, pp. 130–138.
Rights and permissions
About this article
Cite this article
Grigoryan, T.A., Lipacheva, E.V. & Tepoyan, V.H. On the extension of the Toeplitz algebra. Lobachevskii J Math 34, 377–383 (2013). https://doi.org/10.1134/S1995080213040033
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080213040033