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MAXIMAL CHAINS OF CLOSED PRIME IDEALS FOR DISCONTINUOUS ALGEBRA NORMS ON π’ž(K)

J. Esterle
Mathematical Proceedings of the Royal Irish Academy
Vol. 112A, No. 2 (DECEMBER 2012), pp. 101-115
Published by: Royal Irish Academy
Stable URL: http://www.jstor.org/stable/23464472
Page Count: 15
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
MAXIMAL CHAINS OF CLOSED PRIME IDEALS FOR DISCONTINUOUS ALGEBRA NORMS ON π’ž(K)
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Abstract

Let K be an infinite compact space, let π’ž(K) be the algebra of continuous complex-valued functions of K, let 𝓕 be a well-ordered chain of nonmaximal prime ideals of π’ž(K), let π“˜ 𝓕 be the smallest element of 𝓕 and let π“œ 𝓕 be the unique maximal ideal of π’ž(K) containing the elements of 𝓕. Assuming the continuum hypothesis, we show that if $\left|\mathcal{C}\right(\mathrm{K})/{\mathcal{I}}_{\mathcal{F}}|={2}^{{\mathrm{\aleph}}_{0}}$ , and if there exists a sequence (𝓖 n ) nβ‰₯1 of subsets of 𝓕⋃{π“œ 𝓕 } stable under unions such that 𝓕⋃{π“œ 𝓕 } = ⋃ nβ‰₯1 𝓖 n , then there exists a discontinuous algebra norm p on π’ž(K) such that the set of all nonmaximal prime ideals of π’ž(K) which are closed with respect to p equals 𝓕.

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