Quotients of 𝐿^{∞} by Douglas algebras and best approximation

Luecking, Daniel H.; Younis, Rahman M.

doi:10.1090/S0002-9947-1983-0688971-4

Quotients of $L^{\infty }$ by Douglas algebras and best approximation
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by Daniel H. Luecking and Rahman M. Younis PDF

Trans. Amer. Math. Soc. 276 (1983), 699-706 Request permission

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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