Plurisubharmonic Functions and Convexity Properties for General Function Algebras

Abstract

A “natural system” consists of a Hausdorff space $\Sigma$ plus an algebra $\mathfrak {A}$ of complex-valued continuous functions on $\Sigma$ (which contains the constants and determines the topology in $\Sigma$) such that every continuous homomorphism of $\mathfrak {A}$ onto ${\mathbf {C}}$ is given by an evaluation at a point of $\Sigma$ (compact-open topology in $\mathfrak {A}$). The prototype of a natural system is $[{{\mathbf {C}}^n},\mathfrak {P}]$, where $\mathfrak {P}$ is the algebra of polynomials on ${{\mathbf {C}}^n}$. In earlier papers (Pacific J. Math. 18 and Canad. J. Math. 20), the author studied $\mathfrak {A}$-holomorphic functions, which are generalizations of ordinary holomorphic functions in ${{\mathbf {C}}^n}$, and associated concepts of $\mathfrak {A}$-analytic variety and $\mathfrak {A}$-holomorphic convexity in $\Sigma$. In the present paper, a class of extended real-valued functions, called $\mathfrak {A}$-subharmonic functions, is introduced which generalizes the ordinary plurisubharmonic functions in ${{\mathbf {C}}^n}$. These functions enjoy many of the properties associated with plurisubharmonic functions. Furthermore, in terms of the $\mathfrak {A}$-subharmonic functions, a number of convexity properties of ${{\mathbf {C}}^n}$ associated with plurisubharmonic functions can be generalized. For example, if $G$ is an open $\mathfrak {A}$-holomorphically convex subset of $\Sigma$ and $K$ is a compact subset of $G$, then the convex hull of $K$ with respect to the continuous $\mathfrak {A}$-subharmonic functions on $G$ is equal to its hull with respect to the $\mathfrak {A}$-holomorphic functions on $G$.