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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Flows in fibers


Author: Jun-ichi Tanaka
Journal: Trans. Amer. Math. Soc. 343 (1994), 779-804
MSC: Primary 46J15; Secondary 30H05, 54H20
DOI: https://doi.org/10.1090/S0002-9947-1994-1202421-8
MathSciNet review: 1202421
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Abstract: Let $ {H^\infty }(\Delta )$ be the algebra of all bounded analytic functions on the open unit disc $ \Delta $, and let $ \mathfrak{M}({H^\infty }(\Delta ))$ be the maximal ideal space of $ {H^\infty }(\Delta )$. Using a flow, we represent a reasonable portion of a fiber in $ \mathfrak{M}({H^\infty }(\Delta ))$. This indicates a relation between the corona theorem and the individual ergodic theorem. As an application, we answer a question of Forelli [3] by showing that there exists a minimal flow on which the induced uniform algebra is not a Dirichlet algebra. The proof rests on the fact that the closure of a nonhomeomorphic part in $ \mathfrak{M}({H^\infty }(\Delta ))$ may contain a homeomorphic copy of $ \mathfrak{M}({H^\infty }(\Delta ))$. Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1202421-8
Keywords: Maximal ideal spaces, fibers, Gleason parts, minimal flows, Dirichlet algebras
Article copyright: © Copyright 1994 American Mathematical Society

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