More localized automorphisms of the Cuntz algebras

Abstract

AbstractWe completely determine the localized automorphisms of the Cuntz algebras $\mathcal{O}_n$ corresponding to permutation matrices in Mn ⊗ Mn for n = 3 and n = 4. This result is obtained through a combination of general combinatorial techniques and large scale computer calculations. Our analysis proceeds according to the general scheme proposed in a previous paper, where we analysed in detail the case of $\mathcal{O}_2$ using labelled rooted trees. We also discuss those proper endomorphisms of these Cuntz algebras which restrict to automorphisms of their respective diagonals. In the case of $\mathcal{O}_3$ we compute the number of automorphisms of the diagonal induced by permutation matrices in M3 ⊗ M3 ⊗ M3.