Diagonal automorphisms of the $2$-adic ring $C^{\ast }$-algebra

Abstract

The $2$-adic ring $C^{\ast }$-algebra ${\mathcal{Q}}_2$ naturally contains a copy of the Cuntz algebra ${\mathcal{O}}_2$ and, a fortiori, also of its diagonal subalgebra ${\mathcal{D}}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\mathrm{Aut}}_{{\mathcal{D}}_2}({\mathcal{Q}}_2)$ of the automorphisms of ${\mathcal{Q}}_2$ fixing ${\mathcal{D}}_2$ pointwise. It turns out that any such automorphism leaves ${\mathcal{O}}_2$ globally invariant. Furthermore, the subgroup ${\mathrm{Aut}}_{{\mathcal{D}}_2}({\mathcal{Q}}_2)$ is shown to be maximal abelian in $\mathrm{Aut}({\mathcal{Q}}_2)$. Saying exactly what the group is amounts to understanding when an automorphism of ${\mathcal{O}}_2$ that fixes ${\mathcal{D}}_2$ pointwise extends to ${\mathcal{Q}}_2$. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism.

Improved exposition and corrected some typos and inaccuracies

Cited by in corpus (5)

• The inner structure of boundary quotients of right LCM semigroups
• Normalizers and permutative endomorphisms of the $2$-adic ring $C^{\ast }$-algebra
• Permutative representations of the $2$-adic ring $C^{\ast }$-algebra
• On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra
• On the entropy and index of the winding endomorphisms of p-adic ring C${}^{\ast }$-algebras