Plurisubharmonic functions and convexity properties for general function algebras

Abstract

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