Endomorphisms Of Cuntz Algebras Research Articles

A generalization of de Bruijn graphs and classification of endomorphisms of Cuntz algebras by graph invariants

The classification problem of endomorphisms of the Cuntz algebra \(\mathcal{O}_{N}\) is solved by using graph theory. We introduce permutative de Bruijn graphs as generalizations of de Bruijn graphs. Branching laws for a permutative endomorphism ρ of \(\mathcal{O}_{N}\) are computed by using the permutative de Bruijn graph associated with ρ. According to this correspondence between endomorphisms and graphs, we classify permutative endomorphisms of \(\mathcal{O}_{N}\) by graph invariants concretely.

Branching Laws for Polynomial Endomorphisms of Cuntz Algebras Arising from Permutations

In our previous article, we have presented a class of endomorphisms of the Cuntz algebras which are defined by polynomials of canonical generators and their conjugates. We showed the classification of some case under unitary equivalence by help of branching laws of permutative representations which are called cycles. In this article, we generalize results not only the cycle case but also the chain case and we give the maximum of the branching number of such branching laws. Further we classify endomorphisms effectively by graphs associated with their branching laws.