Labeled Trees and Localized Automorphisms of the Cuntz Algebras

More localized automorphisms of the Cuntz algebras

“…However, despite the fact that the Cuntz algebras O n have been intensively studied over the last thirty years, to date precious little was known about the structure of these Weyl groups. In [3], we opened a new and promising line of investigations of this problem. We also discussed at length the case of O 2 therein.…”

“…In [3] we discovered a powerful algorithm to construct those permutations leading to automorphisms; surprisingly, it relies on a certain combinatorial analysis of labeled rooted trees. This fact appears vaguely reminiscent of quantum field theory, although our setup has nothing to do with perturbation theory.…”

“…In this paper, and in [3], we are only concerned with the structure of Aut(O n , D n ) ∩ Aut(O n , F n ), which gives rise (after taking quotient) to the restricted Weyl group.…”

“…where P n = ∪ k P k n , see [3]. Thus, as Cuntz has already shown that every unitary in U(D n ) induces an automorphism of O n , the problem that we are facing is to determine for which permutation matrices w ∈ P k n , k = 1, 2, 3, .…”

“…This is exactly the point where (rooted labeled) trees come to the rescue. For a detailed discussion, see [3]. Throughout the next section, we repeatedly use results from that paper.…”