Polynomial endomorphisms of the Cuntz algebras arising from permutations. I: General theory (Q1777323)

The author introduces and studies a class of endomorphisms of the Cuntz algebras \({\mathcal O}_N\) defined by polynomials of the canonical generators and their adjoints. These endomorphisms \(\psi_\sigma\) are parametrized by the permutations \(\sigma\) of the set \(\{1, \dots,N\}^k\), \(k\geq 1\). It is shown that in a way the endomorphisms \(\psi_\sigma\) behave ``nicely'' with respect to the permutative representations. Using branching laws of permutative representations, a classification of the endomorphisms \(\psi_\sigma\) under unitary equivalence is presented in the particular case when \(N=2\), \(k=2\).