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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Density of parts of algebras on the plane
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by Anthony G. O’Farrell
Trans. Amer. Math. Soc. 196 (1974), 403-414
DOI: https://doi.org/10.1090/S0002-9947-1974-0361795-8

Abstract:

We study the Gleason parts of a uniform algebra A on a compact subset of the plane, where it is assumed that for each point $x \in {\text {C}}$ the functions in A which are analytic in a neighborhood of x are uniformly dense in A. We prove that a part neighborhood N of a nonpeak point x for A satisfies a density condition of Wiener type at $x:\Sigma _{n = 1}^{ + \infty }{2^n}C({A_n}(x)\backslash N) < + \infty$, and if A admits a pth order bounded point derivation at x, then N satisfies a stronger density condition: $\Sigma _{n = 1}^{ + \infty }{2^{(p + 1)n}}C({A_n}(x)\backslash N) < + \infty$. Here C is Newtonian capacity and ${A_n}(x)$ is $\{ z \in {\text {C}}:{2^{ - n - 1}} \leq |z - x| \leq {2^{ - n}}\}$. These results strengthen and extend Browder’s metric density theorem. The relation with potential theory is examined, and analogous results for the algebra ${H^\infty }(U)$ are obtained as corollaries.
References
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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 196 (1974), 403-414
  • MSC: Primary 46J10; Secondary 30A98
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0361795-8
  • MathSciNet review: 0361795