Subharmonic functions and uniform algebras

Abstract

Recently Aupetit and Wermer [J. Functional Anal. <bold>28</bold> (1978), 386-400] have shown conditions under which analytic structure exists in the spectrum space of a uniform algebra. Their work makes critical use of subharmonicity properties of certain classes of functions. In this paper, we develop a technique which offers an alternate and unified proof for subharmonicity of the functions in their paper assuming Basener’s generalized Shilov boundary conjecture. Our technique uses the Oka-Wermer method applied to the <italic>n</italic>-fold tensor product of the algebra. We exhibit further applications of our main result including a special case which holds for all uniform algebras.

Locations

• Proceedings of the American Mathematical Society - View - PDF