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The density of peak points in the Shilov boundary of a Banach function algebra


Author: Taher G. Honary
Journal: Proc. Amer. Math. Soc. 103 (1988), 480-482
MSC: Primary 46J10
DOI: https://doi.org/10.1090/S0002-9939-1988-0943070-9
MathSciNet review: 943070
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Abstract: H. G. Dales has proved in [1] that if $ A$ is a Banach function algebra on a compact metrizable space $ X$, then $ {\bar S_0}(A,X) = \Gamma (A,X)$, where $ {S_0}(A,X)$ is the set of peak points of $ A$ (w.r.t. $ X$) and $ \Gamma (A,X)$ is the Shilov boundary of $ A$ (w.r.t. $ X$). Here, by considering the relation between peak sets and peak points of a Banach function algebra $ A$ and its uniform closure $ \bar A$, we present an elementary and constructive proof of the density of peak points in the Shilov boundary.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0943070-9
Article copyright: © Copyright 1988 American Mathematical Society