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


Quotients of $ L\sp{\infty }$ by Douglas algebras and best approximation

Authors: Daniel H. Luecking and Rahman M. Younis
Journal: Trans. Amer. Math. Soc. 276 (1983), 699-706
MSC: Primary 46J15; Secondary 30H05
MathSciNet review: 688971
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Abstract: We show that $ {L^\infty}/A$ is not the dual space of any Banach space when $ A$ is a Douglas algebra of a certain type. We do this by showing its unit ball has no extreme points. The method used requires that any function in $ {L^\infty}$ has a nonunique best approximation in $ A$. We therefore also show that the Douglas algebra $ {H^\infty} + L_F^\infty $, when $ F$ is an open subset of the unit circle, permits best approximation. We use a method originating in Hayashi [6] and independently obtained by Marshall and Zame.

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PII: S 0002-9947(1983)0688971-4
Keywords: Douglas algebras, dual space, best approximation
Article copyright: © Copyright 1983 American Mathematical Society

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