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Transactions of the American Mathematical Society

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



Splitting of closed ideals in $ ({\rm DFN})$-algebras of entire functions and the property $ ({\rm DN})$

Authors: Reinhold Meise and B. Alan Taylor
Journal: Trans. Amer. Math. Soc. 302 (1987), 341-370
MSC: Primary 32E25; Secondary 32A15, 46J20
MathSciNet review: 887514
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Abstract: For a plurisubharmonic weight function $ p$ on $ {{\mathbf{C}}^n}$ let $ {A_p}({{\mathbf{C}}^n})$ denote the (DFN)-algebra of all entire functions on $ {{\mathbf{C}}^n}$ which do not grow faster than a power of $ \exp (p)$. We prove that the splitting of many finitely generated closed ideals of a certain type in $ {A_p}({{\mathbf{C}}^n})$, the splitting of a weighted $ \overline \partial $-complex related with $ p$, and the linear topological invariant (DN) of the strong dual of $ {A_p}({{\mathbf{C}}^n})$ are equivalent. Moreover, we show that these equivalences can be characterized by convexity properties of $ p$, phrased in terms of greatest plurisubharmonic minorants. For radial weight functions $ p$, this characterization reduces to a covexity property of the inverse of $ p$. Using these criteria, we present a wide range of examples of weights $ p$ for which the equivalences stated above hold and also where they fail.

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Keywords: Algebras of entire functions, slowly decreasing ideals, $ \overline \partial $-operator, structure theory of nuclear Fréchet spaces, linear extension operators, continuous linear right inverse
Article copyright: © Copyright 1987 American Mathematical Society