Geodesic
On the extension of the Toeplitz algebra
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 4, pp. 130-138 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We study the $C^*$-extensions of the Toeplitz algebras with the assistance of isometric operators. It is shown that in the case when the Toeplitz algebra acts irreducibly, all such $C^*$-extensions generate the same algebra, i.e., there is no non-trivial extension of the Toeplitz algebra. We provide the examples of the non-trivial extensions of the Toeplitz algebra in the case when its representation is reducible.
Keywords: inverse semigroup, inverse representation, Toeplitz algebra, $\pi$-extension, inverse $\pi$-extension, $C^*$-algebra. 
@article{UZKU_2012_154_4_a10,
     author = {T. A. Grigoryan and E. V. Lipacheva and V. A. Tepoyan},
     title = {On the extension of the {Toeplitz} algebra},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {130--138},
     year = {2012},
     volume = {154},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2012_154_4_a10/}
}
TY  - JOUR
AU  - T. A. Grigoryan
AU  - E. V. Lipacheva
AU  - V. A. Tepoyan
TI  - On the extension of the Toeplitz algebra
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2012
SP  - 130
EP  - 138
VL  - 154
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/UZKU_2012_154_4_a10/
LA  - ru
ID  - UZKU_2012_154_4_a10
ER  - 
%0 Journal Article
%A T. A. Grigoryan
%A E. V. Lipacheva
%A V. A. Tepoyan
%T On the extension of the Toeplitz algebra
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2012
%P 130-138
%V 154
%N 4
%U http://geodesic.mathdoc.fr/item/UZKU_2012_154_4_a10/
%G ru
%F UZKU_2012_154_4_a10
T. A. Grigoryan; E. V. Lipacheva; V. A. Tepoyan. On the extension of the Toeplitz algebra. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 4, pp. 130-138. http://geodesic.mathdoc.fr/item/UZKU_2012_154_4_a10/

[1] Coburn L. A., “The $C^*$-algebra generated by an isometry”, Bull. Amer. Math. Soc., 73 (1967), 722–726 | DOI | MR | Zbl

[2] Douglas R. G., “On the $C^*$-algebra of a one-parameter semigroup of isometries”, Acta Math., 128:1 (1972), 143–152 | DOI | MR

[3] Murphy G. J., “Ordered groups and Toeplitz algebras”, J. Operator Theory, 18:2 (1987), 303–326 | MR | Zbl

[4] Jang S. Y., “Uniqueness property of $C^*$-algebras like the Toeplitz algebras”, Trends Math., 6:2 (2003), 29–32

[5] Raeburn I., Vittadello S. T., “The isometric representation theory of a perforated semigroup”, J. Operator Theory, 62:2 (2009), 357–370 | MR | Zbl

[6] Grigoryan S. A., Tepoyan V. H., “On isometric representations of the perforated semigroups”, Lobachevskii J. Math., 34:1 (2013), 85–88 | DOI | MR | Zbl

[7] Tepoyan V. H., “On isometric representations of the semigroup $\mathbb Z_+\setminus\{1\}$”, J. Contemp. Math. Anal., 48:2 (2013), 51–57 | DOI | MR

[8] Jang S. Y., “Reduced crossed products by semigroups of automorphisms”, Korean Math. Soc., 36 (1999), 97–107 | MR | Zbl

[9] Grigoryan S. A., Salakhutdinov A. F., “$C^*$-algebry, porozhdennye polugruppami”, Izv. vuzov. Matem., 2009, no. 10, 68–71 | MR | Zbl

[10] Jang S. Y., “Generalized Toeplitz algebras of a certain non-amenable semigroup”, Bull. Korean Math. Soc., 43:2 (2006), 333–341 | DOI | MR

[11] Aukhadiev M. A., Tepoyan V. H., “Isometric representations of totally ordered semigroups”, Lobachevskii J. Math., 33:3 (2012), 39–243 | DOI | MR

[12] Grigoryan S. A., Salakhutdinov A. F., “$C^*$-algebry, porozhdennye polugruppami s sokrascheniem”, Sib. matem. zhurn., 51:1 (2010), 16–25 | MR | Zbl

[13] Klifford A., Preston G., Algebraicheskaya teoriya polugrupp, v 2 t., v. 1, Mir, M., 1972, 286 pp. | Zbl

[14] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984, 469 pp. | MR | Zbl

[15] Arzumanyan V. A., “$*$-predstavleniya inversnykh polugrupp”, Izv. Akademii nauk Arm. SSR, 13:2 (1978), 107–113 | MR | Zbl