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

DOI:
https://doi.org/10.1090/S0002-9947-1972-0317055-2

MathSciNet review:
0317055

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Abstract | References | Similar Articles | Additional Information

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

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0317055-2

Keywords:
Plurisubharmonic functions,
-convexity,
several complex variables,
function algebras,
natural systems

Article copyright:
© Copyright 1972
American Mathematical Society