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Closed ideals of the algebra of absolutely convergent Taylor series


Authors: J. Esterle, E. Strouse and F. Zouakia
Journal: Bull. Amer. Math. Soc. 31 (1994), 39-43
MSC: Primary 43A20; Secondary 46J20, 47A99
MathSciNet review: 1246467
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Abstract: Let $ \Gamma $ be the unit circle, $ A(\Gamma )$ the Wiener algebra of continuous functions whose series of Fourier coefficients are absolutely convergent, and $ {A^ + }$ the subalgebra of $ A(\Gamma )$ of functions whose negative coefficients are zero. If I is a closed ideal of $ {A^ + }$, we denote by $ {S_I}$ the greatest common divisor of the inner factors of the nonzero elements of I and by $ {I^A}$ the closed ideal generated by I in $ A(\Gamma )$. It was conjectured that the equality $ {I^A} = {S_I}{H^{\infty}} \cap {I^A}$ holds for every closed ideal I. We exhibit a large class $ {\mathcal{F}}$ of perfect subsets of $ \Gamma $, including the triadic Cantor set, such that the above equality holds whenever $ h(I) \cap \Gamma \in {\mathcal{F}}$. We also give counterexamples to the conjecture.


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DOI: http://dx.doi.org/10.1090/S0273-0979-1994-00491-4
Article copyright: © Copyright 1994 American Mathematical Society