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Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, Volume 160, Book 2, Pages 275–286 (Mi uzku1452)  

Coverings of solenoids and automorphisms of semigroup $C^*$-algebras

R. N. Gumerov

Kazan Federal University, Kazan, 420008 Russia

Abstract: The paper deals with finite-sheeted covering mappings onto the $P$-adic solenoids and limit endomorphisms of semigroup $C^*$-algebras. The aim of our exposition is two-fold: firstly, to present the results concerning the above-mentioned mappings and endomorphisms; secondly, to demonstrate proofs for some of the results. It has been shown that every covering mapping onto a solenoid is isomorphic to a power mapping. We have considered dynamical properties of the covering mappings. A power mapping for the $P$-adic solenoid is topologically transitive. A criterion for the covering mapping to be chaotic has been given. The classical Euler–Fermat theorem may be used in its proof. We have studied limit endomorphisms of $C^*$-algebras generated by isometric representations for semigroups of rational numbers. We formulate criteria for limit endomorphisms to be automorphisms in number-theoretic, algebraic, and functional terms. The necessity of such a criterion has been given from the category-theoretic viewpoint.

Keywords: automorphism of $C^*$-algebras, chaotic, inductive sequence of Toeplitz algebras associated with sequence of prime numbers, inverse limit and sequence, finite-sheeted covering mapping, semigroup $C^*$-algebra, solenoid, $*$-homomorphism, Toeplitz algebra, topologically transitive.

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Bibliographic databases:
UDC: 517.986
Received: 25.10.2017
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Citation: R. N. Gumerov, “Coverings of solenoids and automorphisms of semigroup $C^*$-algebras”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 160, no. 2, Kazan University, Kazan, 2018, 275–286

Citation in format AMSBIB
\Bibitem{Gum18}
\by R.~N.~Gumerov
\paper Coverings of solenoids and automorphisms of semigroup $C^*$-algebras
\serial Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
\yr 2018
\vol 160
\issue 2
\pages 275--286
\publ Kazan University
\publaddr Kazan
\mathnet{http://mi.mathnet.ru/uzku1452}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3915713}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000460032400008}


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