Density of parts of algebras on the plane

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.