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Closed ideals of the algebra of absolutely convergent Taylor series. (English) Zbl 0803.46052
Summary: 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.

MSC:
46H10 Ideals and subalgebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
46J20 Ideals, maximal ideals, boundaries
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
46F20 Distributions and ultradistributions as boundary values of analytic functions
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