Subharmonic functions and analytic structure in the maximal ideal space of a uniform algebra

In this paper, we present a method of introducing a one-dimensional analytic structure into the maximal ideal space of a uniform algebra, based on the use of subharmonic functions.
Let $A$ be a uniform algebra on a compact Hausdorff space $X$, $M_A$ the maximal ideal space of $A$, $\widehat g$ the Gel'fand transform of a function $g\in A$, $\widehat A=\{\widehat g\mid g\in A\}$, and $p$ a continuous function on $M_A$ which “locally belongs” to the algebra $\widehat A$. In § 1, we introduce certain functions which estimate the “dimensions” of the images of the fibers $p^{-1}(t)$, $t\in p(M_A)$, under mappings $\widehat g$, and prove that these functions are subharmonic. The results obtained on subharmonicity are used to prove the principal theorems of the paper, on finiteness of the fibers and analytic structure (Theorems 1–4). In § 2, these theorems are applied to the study of the maximality properties of algebras of analytic functions on compact subsets of the Riemann sphere.
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