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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Countably generated Douglas algebras


Author: Keiji Izuchi
Journal: Trans. Amer. Math. Soc. 299 (1987), 171-192
MSC: Primary 46J15; Secondary 30D55, 30H05
MathSciNet review: 869406
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Abstract: Under a certain assumption of $ f$ and $ g$ in $ {L^\infty }$ which is considered by Sarason, a strong separation theorem is proved. This is available to study a Douglas algebra $ [{H^\infty },\,f]$ generated by $ {H^\infty }$ and $ f$. It is proved that (1) ball $ (B/{H^\infty } + C)$ does not have exposed points for every Douglas algebra $ B$, (2) Sarason's three functions problem is solved affirmatively, (3) some characterization of $ f$ for which $ [{H^\infty },\,f]$ is singly generated, and (4) the $ M$-ideal conjecture for Douglas algebras is not true.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0869406-9
PII: S 0002-9947(1987)0869406-9
Article copyright: © Copyright 1987 American Mathematical Society