Semicrossed products of the disk algebra

Abstract:

If $\alpha$ is the endomorphism of the disk algebra, $\mathrm {A}(\mathbb {D})$, induced by composition with a finite Blaschke product $b$, then the semicrossed product $\mathrm {A}(\mathbb {D})\times _{\alpha } \mathbb {Z}^+$ imbeds canonically, completely isometrically into $\mathrm {C}(\mathbb {T})\times _{\alpha } \mathbb {Z}^+$. Hence in the case of a non-constant Blaschke product $b$, the C*-envelope has the form $\mathrm {C}(\mathcal {S}_{b}) \times _{s} \mathbb {Z}$, where $(\mathcal {S}_{b}, s)$ is the solenoid system for $(\mathbb {T}, b)$. In the case where $b$ is a constant, the C*-envelope of $\mathrm {A}(\mathbb {D}) \times _{\alpha } \mathbb {Z}^+$ is strongly Morita equivalent to a crossed product of the form $\mathrm {C}_0 (\mathcal S_{e})\times _{s} \mathbb {Z}$, where $e \colon \mathbb {T} \times \mathbb {N} \longrightarrow \mathbb {T} \times \mathbb {N}$ is a suitable map and $(\mathbb {S}_{e}, s)$ is the solenoid system for $(\mathbb {T} \times \mathbb {N}, e)$.