Labeled trees and localized automorphisms of the Cuntz algebras

Abstract

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras O n which preserve both the diagonal and the core U HF -subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of Aut(O n ) and leads to numerous new examples. In particular, we completely classify all such automorphisms of O 2 for the permutation unitaries in ⊗ 4 M 2 . We show that the subgroup of Out(O 2 ) generated by these automorphisms contains a copy of the infinite dihedral group Z ⋊ Z 2 .

Endomorphisms of On which preserve the canonical UHF-subalgebra

“…Thus from now on we make a simplifying assumption that for all (α, β) ∈ J we have |α| − |β| ∈ {−1, 0, +1}. (4). Case k = 1 is established just above this lemma.…”

“…Example 2.5. We give an explicit example of a permutation unitary u ∈ P 4 2 and a unitary v in S 2 \ P 2 such that v ∈ (λ u , λ u ). Indeed, one can check by a lengthy but straightforward computation that the pair…”

“…In order to provide a simple example, we consider the following situation. Let w ∈ P n be such that λ w (D n ) = D n but λ w / ∈ Aut(O n ) (many examples of such permutation unitaries were provided in [4]). Let w ∈ S n \ P n be such that λ w ∈ Aut(O n ) (e.g., an inner one), so that λ w (D n ) = D n .…”

Automorphisms of the Uhf Algebra That Do Not Extend to the Cuntz Algebra

“…In contrast, this last fact is not true for the canonical Cartan subalgebra D n of O n for there are proper endomorphisms of O n that restrict to automorphisms of D n (see [4]). Still, one can show the existence of automorphisms of D n that do not extend to endomorphisms of O n .…”

“…Still, one can show the existence of automorphisms of D n that do not extend to endomorphisms of O n . We conclude this introductory section with a few words on notation (see [4]). For each integer n ≥ 2, the Cuntz algebra O n is the universal C * algebra generated by n isometries S 1 , .…”