$C^*$-algebras generated by mappings. Criterion of irreducibility

S. A. Grigoryan ; A. Yu. Kuznetsova

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2018), pp. 10-22.

Résumé

We study the operator algebra associated with a self-mapping $\varphi $ on a countable set $ X $ which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding $ l^ 2(X) $. We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic $C^*$-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.

Détail

Keywords: $C^*$-algebra, partial isometry, positive operator, projection, compact operator, Toeplitz algebra, extension of $C^*$-algebra by compact operators.

Bibliographie

[1] Murray F. J., von Newmann J., “On rings of operators”, Ann. Math., 37 (1936), 116–229 | DOI | MR

[2] Arveson W., “Operator algebras and measure preserving automorphisms”, Acta Math., 118:1–2 (1967), 95–105 | DOI | MR

[3] Dye H., “On group of measure preserving transformations. I”, Amer. J. Math., 81:1 (1959), 119–159 | DOI | MR | Zbl

[4] Dye H., “On group of measure preserving transformations. II”, Amer. J. Math., 85:4 (1963), 551–576 | DOI | MR | Zbl

[5] Krieger W., “On non-singular transformations of a mesure space. I, II”, Z. Wahr. verw. Geb., 11 (1969), 83–119 | DOI | MR | Zbl

[6] Krieger W., “On ergodic flows and isomorphisms of factors”, Math. Annal., 223 (1976), 19–70 | DOI | MR | Zbl

[7] Arzumanyan V. A., Vershik A. M., “Faktor-predstavleniya skreschennogo proizvedeniya kommutativnoi $C^*$-algebry i polugruppy ee endomorfizmov”, DAN SSSR, 238:3 (1978), 513–516 | Zbl

[8] Arzumanyan V. A., “Operatornye algebry, assotsiirovannye s nesingulyarnymi endomorfizmami prostranstva Lebega”, Izv. AN Armyansk. SSR, 21:6 (1986), 596–616 | MR

[9] Deacony V., “Grouppoids associated with endomorphisms”, Trans. Amer. Math. Soc., 347 (1995), 1779–1786 | DOI | MR

[10] Renault J., “Cuntz-like algebras”, Operator theoretical methods, Proceedings of the 17th international conference on operator theory (Timisoara, Romania, 1998), The Theta Found., Bucharest, 2000, 371–386 | MR | Zbl

[11] Exel R., Vershik A., $C^{*}$-algebras of irreversible dynamical systems, 2002, arXiv: math/0203185v1 [math.OA] | MR

[12] Grigoryan S., Kuznetsova A., “$C^{*}$-algebras generated by mappings”, Lobachevskii J. Math., 29:1 (2008), 5–8 | DOI | MR | Zbl

[13] Grigoryan S. A., Kuznetsova A. Yu., “$C^{*}$-algebry, porozhdennye otobrazheniyami”, Matem. zametki, 87:5 (2010), 694–703 | DOI | Zbl

[14] Cuntz J., “On the simple $C^{*}$-algebras generated by isometries”, Commun. Math. Phys., 57 (1977), 173–185 | DOI | MR | Zbl

[15] Cuntz J., Krieger W., “A class of $C^{*}$-algebras and topological Markov chains”, Invent. Math., 56:3 (1980), 251–268 | DOI | MR | Zbl

[16] Pimsner M. V., “A class of $C^{*}$-algebras generalizing both Cuntz–Krieger algebras and crossed products by $\mathbb{Z}$”, Fields Inst. Commun., 12 (1997), 189–212 | MR | Zbl

[17] Nica A., “$C^{*}$-algebras generated by isometries and Wiener–Hopf operators”, J. Operator Theory, 27 (1992), 17–52 | MR | Zbl

[18] Matsumoto K., “Relation among generators of $C^{*}$-algebras associated with subshifts”, Internat. J. Math., 10 (1999), 385–405 | DOI | MR | Zbl

[19] Exel R., Laca M., Quigg J., Partial dynamical systems and $C^{*}$-algebras generated by partial isometries, arXiv: funct-an/9712007

[20] Kuznetsova A. Yu., “Ob odnom klasse operatornykh algebr, porozhdennykh semeistvom chastichnykh izometrii”, Zap. nauchn. semin. POMI, 437, 2015, 131–144

[21] Grigoryan S. A., Kuznetsova A. Yu., Patrin E. V., “Ob odnom kriterii neprivodimosti algebry $C^*_{\varphi}(X)$”, Izv. NAN Armenii. Matem., 49:1 (2014), 75–82

[22] Grigoryan S., Kuznetsova A., “On a class of nuclear $C^{*}$-algebras”, An operator theory summer, Proceedings of the 23rd international conference on operator theory (Timisoara, Romania, 2010), The Theta Found., Bucharest, 2012, 39–50 | MR

[23] Grigoryan S., Kuznetsova A., “Torus action on $C^{*}$-algebra”, The varied landscape of operator theory, Proceedings of the 24th international conference on operator theory (Timisoara, Romania, 2012), The Theta Found., Bucharest, 2014, 127–136 | MR

[24] Kuznetsova A. Yu., “Ob odnom klasse $C^{*}$-algebr, porozhdennykh schetnym semeistvom chastichnykh izometrii”, Izv. NAN Armenii. Matem., 45:6 (2010), 51–62 | MR | Zbl

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

[26] Coburn L., “The $C^{*}$-algebra generated by an isometry. II”, Trans. Amer. Math. Soc., 137 (1969), 211–217 | MR | Zbl