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# MAXIMAL CHAINS OF CLOSED PRIME IDEALS FOR DISCONTINUOUS ALGEBRA NORMS ON 𝒞(K)

J. Esterle
Mathematical Proceedings of the Royal Irish Academy
Vol. 112A, No. 2 (DECEMBER 2012), pp. 101-115
Stable URL: http://www.jstor.org/stable/23464472
Page Count: 15
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## Abstract

Let K be an infinite compact space, let 𝒞(K) be the algebra of continuous complex-valued functions of K, let 𝓕 be a well-ordered chain of nonmaximal prime ideals of 𝒞(K), let 𝓘 𝓕 be the smallest element of 𝓕 and let 𝓜 𝓕 be the unique maximal ideal of 𝒞(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 ) n≥1 of subsets of 𝓕⋃{𝓜 𝓕 } stable under unions such that 𝓕⋃{𝓜 𝓕 } = ⋃ n≥1 𝓖 n , then there exists a discontinuous algebra norm p on 𝒞(K) such that the set of all nonmaximal prime ideals of 𝒞(K) which are closed with respect to p equals 𝓕.

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