A tetrachotomy for certain algebras containing the disc algebra

Izzo, Alexander

doi:10.1090/S0002-9939-09-10097-7

A tetrachotomy for certain algebras containing the disc algebra
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by Alexander J. Izzo PDF

Proc. Amer. Math. Soc. 138 (2010), 623-627 Request permission

Abstract:

Let $D$ denote the open unit disc in the complex plane, and let $A(D)$ denote the disc algebra. The algebras $A(D)[f, \overline f ] \cap C(\overline D)$, where $f$ is in $H^\infty (D)$, are characterized. This answers a question raised by Joseph Cima.

References

Sheldon Axler and Allen Shields, Algebras generated by analytic and harmonic functions, Indiana Univ. Math. J. 36 (1987), no. 3, 631–638. MR 905614, DOI 10.1512/iumj.1987.36.36034
E. M. Čirka, Approximation by holomorphic functions on smooth manifolds in $\textbf {C}^{n}$, Mat. Sb. (N.S.) 78 (120) (1969), 101–123 (Russian). MR 0239121
T. W. Gamelin, Uniform Algebras, 2nd ed., Chelsea, New York, 1984.
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
Alexander J. Izzo, Algebras containing bounded holomorphic functions, Indiana Univ. Math. J. 52 (2003), no. 5, 1305–1342. MR 2010729, DOI 10.1512/iumj.2003.52.2315
Alexander J. Izzo, Algebras generated by the disc algebra and bounded harmonic functions, Proc. Amer. Math. Soc. 135 (2007), no. 4, 1065–1071. MR 2262907, DOI 10.1090/S0002-9939-06-08547-9
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157