Coverings of solenoids and automorphisms of semigroup C*-algebras

R.N. Gumerov

Kazan Federal University, Kazan, 420008 Russia

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Abstract

The paper deals with finite-sheeted covering mappings onto the C*-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 C*-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, C*-homomorphism, Toeplitz algebra, topologically transitive

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Received

October 25, 2017

For citation: Gumerov R.N. Coverings of solenoids and automorphisms of semigroup C*-algebras. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 275–286. 

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