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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Algebras generated by the disc algebra and bounded harmonic functions


Author: Alexander J. Izzo
Journal: Proc. Amer. Math. Soc. 135 (2007), 1065-1071
MSC (2000): Primary 46J10, 46J15, 30H05
Published electronically: September 26, 2006
MathSciNet review: 2262907
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Abstract: Let $ D$ denote the open unit disc, and let $ A(D)$ denote the disc algebra. The subsets $ E$ of $ \partial D$ such that the inclusion $ A(D)[f,{\overline f}]\supset C(\overline D)$ holds for every nonconstant $ f\in H^\infty (D)$ continuous on $ E$, or the inclusion $ A(D)[f] \supset C(\overline D)$ holds for every bounded harmonic nonholomorphic function $ f$ on $ D$ continuous on $ E$, are characterized. In the first case the condition is that $ E$ has positive measure, and in the second case that $ E$ has full measure in $ \partial D$.


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Additional Information

Alexander J. Izzo
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
Email: aizzo@math.bgsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08547-9
PII: S 0002-9939(06)08547-9
Received by editor(s): January 12, 2005
Received by editor(s) in revised form: November 1, 2005
Published electronically: September 26, 2006
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2006 American Mathematical Society