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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Plurisubharmonic functions and convexity properties for general function algebras
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by C. E. Rickart PDF
Trans. Amer. Math. Soc. 169 (1972), 1-24 Request permission

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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 169 (1972), 1-24
  • MSC: Primary 46J10; Secondary 32F05, 46G20
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0317055-2
  • MathSciNet review: 0317055