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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Minimal superalgebras of weak-$ \sp \ast$ Dirichlet algebras


Author: Takahiko Nakazi
Journal: Proc. Amer. Math. Soc. 95 (1985), 70-72
MSC: Primary 46J10
DOI: https://doi.org/10.1090/S0002-9939-1985-0796448-8
MathSciNet review: 796448
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Abstract: Let $ A$ be a weak-$ * $ Dirichlet algebra in $ {L^\infty }(m)$ and let $ {H^\infty }(m)$ be the weak-$ * $ closure of $ A$ in $ {L^\infty }(m)$. It may happen that there are minimal weak-$ * $ closed subalgebras of $ {L^\infty }(m)$ that contain $ {H^\infty }(m)$ properly. In this paper it is shown that if there is a minimal, proper, weak-$ * $ closed superalgebra of $ {H^\infty }(m)$, then, in fact, that algebra is the unique least element in the lattice of all proper weak-$ * $ closed superalgebras of $ {H^\infty }(m)$.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0796448-8
Keywords: Minimal weak-$ * $ closed superalgebras, weak-$ * $ Dirichlet algebra, Hardy space, uniform algebra
Article copyright: © Copyright 1985 American Mathematical Society