Normalizers and permutative endomorphisms of the $2$-adic ring $C^{\ast }$-algebra

Abstract

A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra ${\mathcal{D}}_2$ in the $2$-adic ring $C^{\ast }$-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 $U$.

To appear in J. Math. Anal. Appl