Maximality and symmetry related to the $2$-adic ring $C^{\ast }$-algebra

The 2-adic ring $C^{\ast }$-algebra ${\mathcal{Q}}_2$ is the universal $C^{\ast }$-algebra generated by a unitary and an isometry satisfying certain relations. It contains a canonical copy of the Cuntz algebra ${\mathcal{O}}_2$. We show that ${\mathcal{O}}_2$ is a maximal $C^{\ast }$-subalgebra of ${\mathcal{Q}}_2$. Furthermore, we examine the structure of the fixed-point algebra under a periodic ${}^{\ast }$-automorphism $\sigma$ of ${\mathcal{Q}}_2$, which is extended from the flip-flop ${}^{\ast }$-automorphism of ${\mathcal{O}}_2$. We show that the maximality of ${\mathcal{O}}_2$ in ${\mathcal{Q}}_2$ extends to the crossed product ${\mathcal{O}}_2{⋊}_{\sigma }{\mathbb{Z}}_2$ in ${\mathcal{Q}}_2{⋊}_{\sigma }{\mathbb{Z}}_2$, and to the fixed-point algebra ${\mathcal{O}}_2^{\sigma }$ in ${\mathcal{Q}}_2^{\sigma }$. As a consequences of our main results, a few open questions concerning ${\mathcal{Q}}_2$ are resolved.