# Normalizers and permutative endomorphisms of the $2$-adic ring $C^*$-algebra

## TL;DR

This paper characterizes the unitary normalizer of the diagonal subalgebra in the 2-adic ring C*-algebra, explores the non-regularity of a certain inclusion, and introduces new permutative endomorphisms with specific properties.

## Contribution

It provides a comprehensive description of the normalizer, proves the non-regularity of the inclusion, and constructs novel permutative endomorphisms of the 2-adic ring C*-algebra.

## Key findings

- Complete description of the unitary normalizer of the diagonal subalgebra.
- Proof that the inclusion 2  in 2 is not regular.
- Construction of countably many new permutative endomorphisms with prescribed images.

## Abstract

A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra $\mathcal{D}_2$ in the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$, which generalizes and unifies analogous results for Cuntz and Bunce-Deddens algebras. Furthermore, the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$ is proved not to be regular. Finally, countably many novel permutative endomorphisms of $\mathcal{Q}_2$ are exhibited with prescribed images of the generator $U$.