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 Izv. Vyssh. Uchebn. Zaved. Mat., 2018, Number 2, Pages 10–22 (Mi ivm9326)

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

S. A. Grigoryana, A. Yu. Kuznetsovab

a Kazan State Energy University, 51 Krasnoselskaya str., Kazan, 420066 Russia
b Kazan Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia

Abstract: 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.

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

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2018, 62:2, 7–18

Bibliographic databases:

UDC: 517.988:519.3

Citation: S. A. Grigoryan, A. Yu. Kuznetsova, “$C^*$-algebras generated by mappings. Criterion of irreducibility”, Izv. Vyssh. Uchebn. Zaved. Mat., 2018, no. 2, 10–22; Russian Math. (Iz. VUZ), 62:2 (2018), 7–18

Citation in format AMSBIB
\Bibitem{GriKuz18} \by S.~A.~Grigoryan, A.~Yu.~Kuznetsova \paper $C^*$-algebras generated by mappings. Criterion of irreducibility \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2018 \issue 2 \pages 10--22 \mathnet{http://mi.mathnet.ru/ivm9326} \transl \jour Russian Math. (Iz. VUZ) \yr 2018 \vol 62 \issue 2 \pages 7--18 \crossref{https://doi.org/10.3103/S1066369X18020020} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000427510500002} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85043984575} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. A. Grigoryan, A. Yu. Kuznetsova, “$C^*$-algebras generated by mappings. Classification of invariant subspaces”, Russian Math. (Iz. VUZ), 62:7 (2018), 13–30
2. A. Yu. Kuznetsova, “Algebra associated with a map inducing an inverse semigroup”, Lobachevskii J. Math., 40:8, SI (2019), 1102–1112
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