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Proceedings of the American Mathematical Society

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Kahane's construction and the weak sequential completeness of $ L\sp{1}$


Author: E. A. Heard
Journal: Proc. Amer. Math. Soc. 44 (1974), 96-100
MSC: Primary 46E30
DOI: https://doi.org/10.1090/S0002-9939-1974-0365115-X
MathSciNet review: 0365115
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Abstract: If $ \lambda $ is a normalized Lebesgue measure $ ($arc length$ /2\pi )$ on $ T = \{ \vert z\vert = 1\} $, the Gelfand map permits $ {L^\infty }{(\lambda )^ \ast }$ to be identified with $ M(X)$, the space of finite Baire measures on $ X$, the maximal ideal space of $ {L^\infty }(\lambda )$. The measure $ {m_0}$ in $ M(X)$ represents $ \lambda :\int {X\hat fd{m_0} = \int {Tfd\lambda } } $ for all $ f \in {L^\infty }(\lambda )$. Furthermore $ \mu \ll {m_0}$ if and only if $ \mu $ represents some measure of the form $ \phi d\lambda ,\phi \in {L^1}(\lambda )$. Using this fact and a sum constructed by J. P. Kahane, $ \Sigma {\hat h^{n(j)}}$ when $ \hat h$ is an appropriate function guaranteed by Urysohn's lemma, develops a proof that $ {L^1}(\lambda )$ is weakly sequentially complete.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0365115-X
Article copyright: © Copyright 1974 American Mathematical Society