The Bergman-Shilov boundary for subfamilies of $q$-plurisubharmonic functions

Abstract

We introduce a notion of the Bergman-Shilov (or Shilov) boundary for some subclasses 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 subclasses with simple structure one can show the existence and uniqueness of the Shilov boundary. Then 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. In the case of bounded pesudoconvex domains with smooth boundary we also show that some parts of the Shilov boundary for $q$-plurisubharmonic functions are foliated by $q$-dimensional complex submanifolds.