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Semicrossed products of the disk algebra


Authors: Kenneth R. Davidson and Elias G. Katsoulis
Journal: Proc. Amer. Math. Soc. 140 (2012), 3479-3484
MSC (2000): Primary 47L55
DOI: https://doi.org/10.1090/S0002-9939-2012-11348-6
Published electronically: February 17, 2012
MathSciNet review: 2929016
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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)$.


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Additional Information

Kenneth R. Davidson
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L–3G1, Canada
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.
Email: katsoulise@ecu.edu

DOI: https://doi.org/10.1090/S0002-9939-2012-11348-6
Keywords: Semicrossed product, crossed product, disk algebra, C*-envelope
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
Article copyright: © Copyright 2012 American Mathematical Society

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