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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, $ p$-convexity, several complex variables, function algebras, natural systems
Article copyright: © Copyright 1972 American Mathematical Society