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Abstract

We introduce the notion of the Shilov boundary for some subfamilies of upper semicontinuous functions on a compact Hausdorff space. It is by definition the smallest closed subset of the given space on which all functions of that subclass attain their maximum. For certain subfamilies with simple structure we show the existence and uniqueness of the Shilov boundary. We provide its relation to the set of peak points and establish Bishop-type theorems. As an application we obtain a generalization of Bychkov’s theorem which gives a geometric characterization of the Shilov boundary for $q$-plurisubharmonic functions on convex bounded domains.