Analytic geometry on compacta in \(\mathbb{C}^ n\)
We consider new objects: cluster of perfect sequences of holomorphic mappings of the unit disk \(U\) into \(\mathbb{C}\). Each clusters of perfect sequences of holomorphic mappings of the unit disk \(U\) into \(\mathbb{C}^n\). Each cluster bears a measure on its boundary which we call the holomorphic measure. We show that clusters and measures form a compact convex set and enjoy many properties characteristic for disks on the plane. We prove that a function \(u\) is psh if and only if its restriction to such clusters satisfies the subaveraging property with respect to holomorphic measures. In particular, if there is no clusters of perfect sequences in \(X\) then every continuous function on \(X\) is plurisubharmonic. Because of the apparent importance of perfect sequences, it is reasonable to study their geometry, which we call the analytic geometry of compacta. This geometry is invariant under biholomorphic transformations and the basic picture has the following features: let \(O_X\) be the set of peak points for the cone of plurisubharmonic functions on a compact \(X\subset C^n\) and let \(B_X\) be the closure of \(O_X\). Then through every point \(z\) of \(X\) passes the cluster of a perfect sequence with its boundary belonging to \(B_X\). Moreover, there is the maximal possible cluster passing through \(z\). The picture, of course, is more detailed and some of these details are presented in this paper. We apply our results to uniform algebras of holomorphic functions on compacta. First, we show that a point \(z\) is the psh hull of a compact \(X\) if and only if there is the cluster of a perfect sequence centered at \(z\) with its boundary belonging to \(X\). Then we show that clusters of perfect sequences provide the cover of \(X\) such that algebras of holomorphic functions on elements of this cover are weakly analytic. We also find a cover of \(X\) by sets with algebras of holomorphic functions analytic at centers of these sets. We offer a simple criterion for analyticity of such algebras on clusters of perfect sequences which allows us to construct a compact of Wermer's type with analytic algebra of holomorphic functions.
scientific article; zbMATH DE number 838470