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
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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