Automorphisms of the limits for the direct sequences of the Toeplitz-Cuntz algebras

The Toeplitz-Cuntz algebras are universal $C^*$-algebras generated by finite families of isometries with pairwise orthogonal ranges. We consider the direct sequences of the Toeplitz-Cuntz algebras whose connecting homomorphisms are defined by tuples consisting of sequences of natural numbers. The limits of such direct sequences are isomorphic to the reduced semigroup $C^*$-algebras for free products of semigroups of positive rational numbers.
In the talk, we construct morphisms between two copies of the same direct sequence of the Toeplitz-Cuntz algebras. These morphisms induce the limit $*$-endomorphisms of the $C^*$-algebra that is the direct limit for this direct sequence. We establish the criterion for the limit $*$-endomorphisms to be automorphisms of the direct limits.