# Diagonal automorphisms of the $2$-adic ring $C^*$-algebra

## TL;DR

This paper characterizes automorphisms of the 2-adic ring C*-algebra that fix the diagonal subalgebra, showing they preserve the Cuntz algebra and are composed of localized inner and gauge automorphisms.

## Contribution

It provides a complete description of automorphisms fixing the diagonal in the 2-adic ring C*-algebra, especially for localized automorphisms.

## Key findings

- Automorphisms fixing the diagonal preserve the Cuntz algebra.
- The subgroup fixing the diagonal is maximal abelian.
- Localized automorphisms extend iff they are composed of localized inner and gauge automorphisms.

## Abstract

The $2$-adic ring $C^*$-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 ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ 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 ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shown to be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. 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.