Abstract
The paper deals with the direct sequences of the Toeplitz-Cuntz algebras whose connecting homomorphisms are defined by tuples consisting of sequences of prime numbers. We study properties of limit ⁎-endomorphisms of the C⁎-algebras that are the direct limits for such direct sequences. The limit ⁎-endomorphisms are induced by morphisms between copies of the same direct sequences of the Toeplitz-Cuntz algebras. We establish the criteria for the limit ⁎-endomorphisms to be automorphisms of the direct limits. These criteria are formulated in number-theoretic, algebraic and functional terms. In particular, it is shown that a limit ⁎-endomorphism is an automorphism if and only if there exist generalized means on the P-adic solenoids.
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