The Forelli problem concerning ideals in the disk algebra $A(\textbf {D})$
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- by Raymond Mortini PDF
- Proc. Amer. Math. Soc. 95 (1985), 261-264 Request permission
Abstract:
Let $Z(f)$ be the zero set of a function $f \in A({\mathbf {D}})$ and $Z(I) = { \cap _{f \in I}}Z(f)$ the zero set of an ideal $I$ in $A({\mathbf {D}})$. It is shown that in the disk algebra $A({\mathbf {D}})$ every finitely generated ideal $I$ has the weak Forelli property, i.e. there exists a function $f \in I$ such that $Z(f) \cap T = Z(I) \cap T$, where $T$ is the boundary of the unit circle ${\mathbf {D}}$. On the other hand, there exists a finitely generated ideal $I$ in $A({\mathbf {D}})$ such that $Z(f) \ne Z(I)$ for each choice of $f \in I$. This provides us with a negative answer to a problem of F. Forelli [1].References
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- Leonard Gillman and Meyer Jerison, Rings of continuous functions, Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition. MR 0407579
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Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 261-264
- MSC: Primary 46J15; Secondary 30H05
- DOI: https://doi.org/10.1090/S0002-9939-1985-0801335-2
- MathSciNet review: 801335