Semicrossed products of the disc algebra

If $\alpha$ is the endomorphism of the disk algebra, $\text{AD}$, induced by composition with a finite Blaschke product $b$, then the semicrossed product $\text{AD}{\times }_{\alpha }{\text{bZ}}^{+}$ imbeds canonically, completely isometrically into $\text{rC}(\text{bT}){\times }_{\alpha }{\text{bZ}}^{+}$. Hence in the case of a non-constant Blaschke product $b$, the C*-envelope has the form $\text{rC}({\mathrm{\S }}_b){\times }_s\text{bZ}$, where $({\mathrm{\S }}_b,s)$ is the solenoid system for $(\text{bT},b)$. In the case where $b$ is a constant, then the C*-envelope of $\text{AD}{\times }_{\alpha }{\text{bZ}}^{+}$ is strongly Morita equivalent to a crossed product of the form $\text{rC}({\mathrm{\S }}_e){\times }_s\text{bZ}$, where $e\text{\hspace{0.17em}⁣}:\text{bT}\times \text{bN}⟶\text{bT}\times \text{bN}$ is a suitable map and $({\mathrm{\S }}_e,s)$ is the solenoid system for $(\text{bT}\times \text{bN},e)$ .