Maximality and symmetry related to the \(2\)-adic ring \(C^*\)-algebra

The 2-adic ring $C^*$-algebra $\mathcal{Q}_2$ is the universal $C^*$-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 $C^*$-subalgebra of $\mathcal{Q}_2$. Furthermore, we examine the structure of the fixed-point algebra under a periodic $^*$-automorphism $\sigma$ of $\mathcal{Q}_2$, which is extended from the flip-flop $^*$-automorphism of $\mathcal{O}_2$. We show that the maximality of $\mathcal{O}_2$ in $\mathcal{Q}_2$ extends to the crossed product $\mathcal{O}_2 \rtimes_{\sigma} \mathbb{Z}_2$ in $\mathcal{Q}_2 \rtimes_{\sigma} \mathbb{Z}_2$, 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.