Semicrossed products of the disk algebra
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- by Kenneth R. Davidson and Elias G. Katsoulis PDF
- Proc. Amer. Math. Soc. 140 (2012), 3479-3484 Request permission
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 $(\mathbb {T} \times \mathbb {N}, e)$.References
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Additional Information
- Kenneth R. Davidson
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2Lā3G1, Canada
- MR Author ID: 55000
- ORCID: 0000-0002-5247-5548
- Email: krdavids@uwaterloo.ca
- Elias G. Katsoulis
- Affiliation: Department of Mathematics, University of Athens, 15784 Athens, Greece
- Address at time of publication: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858.
- MR Author ID: 99165
- Email: katsoulise@ecu.edu
- Received by editor(s): April 7, 2011
- Published electronically: February 17, 2012
- Additional Notes: The first author was partially supported by an NSERC grant.
The second author was partially supported by a grant from the ECU - Communicated by: Marius Junge
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3479-3484
- MSC (2000): Primary 47L55
- DOI: https://doi.org/10.1090/S0002-9939-2012-11348-6
- MathSciNet review: 2929016