绝对收敛Taylor级数代数的闭理想
Closed ideals of the algebra of absolutely convergent Taylor series
摘要：
设$Γ$为单位圆，$A(Γ)$为Fourier系数级数绝对收敛的连续函数的Wiener代数，$A^+$为负系数为零的函数的$A(Γ)$的子代数.若$I$是$A^+$的闭理想，则用$S_I$表示$I$的非零元素的内因子的最大公约数，用$I^A$表示$I$在$A(Γ)$中生成的闭理想.证明了等式$I^A= S_I H^\infty \cap I^A$对每个闭理想$I$成立.我们展示了一个大类$\scr F$完美的$Γ$的子集，包括三元康托集，使上述等式持有时，$h(I)\capΓ\in\scr F$。我们还给出了该猜想的反例。
英文摘要：
Let $Γ$ be the unit circle, $A(Γ)$ the Wiener algebra of continuous functions whose series of Fourier coefficients are absolutely convergent, and $A^+$ the subalgebra of $A(Γ)$ 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 $A(Γ)$. 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 $\scr F$ of perfect subsets of $Γ$, including the triadic Cantor set, such that the above equality holds whenever $h(I)\capΓ\in\scr F$. We also give counterexamples to the conjecture.
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