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 Mat. Zametki, 2010, Volume 87, Issue 5, Pages 694–703 (Mi mz3884)

$C^*$-Algebras Generated by Mappings

S. A. Grigoryana, A. Yu. Kuznetsovab

a Kazan State Power Engineering University
b Kazan State University

Abstract: In the paper, some properties of a singly generated $C^*$-subalgebra of the algebra of all bounded operators $B(l^2(X))$ on the Hilbert space $l^2(X)$ with the generator $T_\varphi$ induced by a mapping $\varphi$ of an infinite set $X$ into itself are investigated. A condition on $\varphi$ is presented under which the operator $T_\varphi$ is continuous, and it is proved that, if this is the case, then the study of these algebras can be reduced to that of $C^*$-algebras generated by a finite family of partial isometries of a special form. A complete description of the $C^*$-algebras generated by an injective mapping on $X$ is given. Examples of $C^*$-algebras generated by noninjective mappings are treated.

Keywords: C^*$-algebra,$C^*$-algebra generated by a mapping, injective mapping, partial isometry, Toeplitz algebra, Cuntz algebra DOI: https://doi.org/10.4213/mzm3884 Full text: PDF file (501 kB) References: PDF file HTML file English version: Mathematical Notes, 2010, 87:5, 663–671 Bibliographic databases: UDC: 517.986.32 Received: 22.08.2006 Revised: 30.01.2007 Citation: S. A. Grigoryan, A. Yu. Kuznetsova, “$C^*$-Algebras Generated by Mappings”, Mat. Zametki, 87:5 (2010), 694–703; Math. Notes, 87:5 (2010), 663–671 Citation in format AMSBIB \Bibitem{GriKuz10} \by S.~A.~Grigoryan, A.~Yu.~Kuznetsova \paper$C^*$-Algebras Generated by Mappings \jour Mat. Zametki \yr 2010 \vol 87 \issue 5 \pages 694--703 \mathnet{http://mi.mathnet.ru/mz3884} \crossref{https://doi.org/10.4213/mzm3884} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2766584} \transl \jour Math. Notes \yr 2010 \vol 87 \issue 5 \pages 663--671 \crossref{https://doi.org/10.1134/S0001434610050068} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000279600700006} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77954420707}  Linking options: • http://mi.mathnet.ru/eng/mz3884 • https://doi.org/10.4213/mzm3884 • http://mi.mathnet.ru/eng/mz/v87/i5/p694  SHARE: Citing articles on Google Scholar: Russian citations, English citations Related articles on Google Scholar: Russian articles, English articles This publication is cited in the following articles: 1. M. A. Aukhadiev, S. A. Grigoryan, E. V. Lipacheva, “A compact quantum semialgebra generated by an isometry”, Russian Math. (Iz. VUZ), 55:10 (2011), 78–81 2. A. Yu. Kuznetsova, E. V. Patrin, “One class of$C^*$-algebras generated by a family of partial isometries and multiplicators”, Russian Math. (Iz. VUZ), 56:6 (2012), 37–47 3. A. Yu. Kuznetsova, E. V. Patrin, “Ob idealakh$C^*$-algebry, porozhdennoi semeistvom chastichnykh izometrii i multiplikatorami”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 157, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2015, 51–59 4. A. Yu. Kuznetsova, “On a class of operator algebras generated by a family of partial isometries”, J. Math. Sci. (N. Y.), 216:1 (2016), 84–93 5. E. V. Patrin, “O graduirovkakh$C^*$-algebry, porozhdennoi otobrazheniem i algebroi multiplikatorov”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 157, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2015, 56–66 6. S. A. Grigoryan, A. Yu. Kuznetsova, “$C^*$-algebras generated by mappings. Criterion of irreducibility”, Russian Math. (Iz. VUZ), 62:2 (2018), 7–18 7. S. A. Grigoryan, A. Yu. Kuznetsova, “$C^*$-algebras generated by mappings. Classification of invariant subspaces”, Russian Math. (Iz. VUZ), 62:7 (2018), 13–30 8. Kuznetsova A.Yu., “Algebra Associated With a Map Inducing An Inverse Semigroup”, Lobachevskii J. Math., 40:8, SI (2019), 1102–1112 9. G. G. Amosov, S. A. Grigoryan, A. Yu. Kuznetsova, “O$C^*\$-algebrakh, porozhdennykh mnozhestvom raspredelenii veroyatnostei”, Izv. vuzov. Matem., 2021, no. 8, 66–71
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