The Forelli problem concerning ideals in the disk algebra

Author:
Raymond Mortini

Journal:
Proc. Amer. Math. Soc. **95** (1985), 261-264

MSC:
Primary 46J15; Secondary 30H05

MathSciNet review:
801335

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Abstract: Let be the zero set of a function and the zero set of an ideal in . It is shown that in the disk algebra every finitely generated ideal has the weak Forelli property, i.e. there exists a function such that , where is the boundary of the unit circle . On the other hand, there exists a finitely generated ideal in such that for each choice of . This provides us with a negative answer to a problem of F. Forelli [**1**].

**[1]**Frank Forelli,*A note on ideals in the disc algebra*, Proc. Amer. Math. Soc.**84**(1982), no. 3, 389–392. MR**640238**, 10.1090/S0002-9939-1982-0640238-0**[2]**John B. Garnett,*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971****[3]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition; Graduate Texts in Mathematics, No. 43. MR**0407579****[4]**Kenneth Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008****[5]**Michael von Renteln,*Every subring 𝑅 of 𝑁 with 𝐴(\overline𝐷)⊂𝑅 is not adequate*, Acta Sci. Math. (Szeged)**39**(1977), no. 1-2, 139–140. MR**0447598**

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DOI:
https://doi.org/10.1090/S0002-9939-1985-0801335-2

Article copyright:
© Copyright 1985
American Mathematical Society