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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Plurisubharmonic functions and convexity properties for general function algebras

Author: C. E. Rickart
Journal: Trans. Amer. Math. Soc. 169 (1972), 1-24
MSC: Primary 46J10; Secondary 32F05, 46G20
MathSciNet review: 0317055
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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 $G$.

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Keywords: Plurisubharmonic functions, <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$p$">-convexity, several complex variables, function algebras, natural systems
Article copyright: © Copyright 1972 American Mathematical Society