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

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

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]**F. Forelli,*A note on the ideals in the disk algebra*, Proc. Amer. Math. Soc.**84**(1982), 389-392. MR**83c**46041. MR**640238 (83c:46041)****[2]**J. Garnett,*Bounded analytic functions*, Academic Press, New York, 1981. MR**83g**30037 MR**628971 (83g:30037)****[3]**L. Gillman and M. Jerison,*Rings of continuous functions*, Graduate Texts in Math., No. 43, Springer-Verlag, Berlin and New York, 1976. MR**53**#11352. MR**0407579 (53:11352)****[4]**K. Hoffman,*Banach spaces of analytic functions*, Prentice Hall, Englewood Cliffs, N. J., 1962. MR**24**#A 2844. MR**0133008 (24:A2844)****[5]**M. v. Renteln,*Every subring**of**with**is not adequate*, Acta Sci. Math.**39**(1977), 139-140. MR**56**#5908. MR**0447598 (56:5908)**

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

Article copyright:
© Copyright 1985
American Mathematical Society