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Transactions of the American Mathematical Society

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



Density of parts of algebras on the plane

Author: Anthony G. O’Farrell
Journal: Trans. Amer. Math. Soc. 196 (1974), 403-414
MSC: Primary 46J10; Secondary 30A98
MathSciNet review: 0361795
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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 \vert z - x\vert \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.

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Keywords: Gleason part, metric density, T-invariant algebra
Article copyright: © Copyright 1974 American Mathematical Society