Coverings of solenoids and automorphisms of semigroup C*-algebras

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