On Limit Automorphisms of $$C^{*}$$-Algebras

The article is concerned with endomorphisms for the inductive limits of inductive sequences consisting of $$C^{*}$$ -algebras and their morphisms. These $$C^{*}$$ -algebras are the Toeplitz–Cuntz algebras. For such an inductive sequence, a countable collection of connecting $$\ast$$ -homomorphisms is determined by a finite tuple consisting of infinite sequences of arbitrary natural numbers. We show that for studying properties of the endomorphisms one can use the inductive sequences of the $$C^{*}$$ -algebras with collections of connecting morphisms which arise from finite tuples of sequences of primes. Using this fact and the results of our previous work, we characterize the endomorphisms that are the automorphisms of the inductive limits. This characterization is given in terms of the sequences of natural numbers involved in the construction of connecting morphisms of the inductive sequences of $$C^{*}$$ -algebras. It is also shown that the characterization is closely connected with conditions for constructing generalized means and finite-sheeted connected coverings of topological groups.