Abstract
Let A be a uniform algebra with maximal ideal space M A . A notion of subharmonicity is defined for functions on M A . Under certain hypotheses of continuity, it is proved that the notion of subharmonicity is local. A consequence is that the notion of Jensen boundary point is local. The solutions to an abstract Dirichlet problem are studied in the context of uniform algebras. The methods are applied to algebras of analytic functions, and in particular a version of the extended maximum principle is obtained for analytic functions of several complex variables.
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