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Point derivations and prime ideals in \(R(X)\). (English) Zbl 0742.46033

Using a modification of earlier constructions by R. McKissik [Bull. Am. Math. Soc. 69, 391–395 (1963; Zbl 0113.31502)], J. Li-Ming Wang [Proc. Am. Math. Soc. 51, 141–142 (1975; Zbl 0276.46023)], and J. Wermer [J. Funct. Anal. 1, 28–36 (1967; Zbl 0159.42102)], the author produces various new examples of Swiss cheeses with exotic properties, e.g., he constructs a Swiss cheese \(X\) such that \(0\in X\), \(R(X)\) has a non-zero continuous point derivation at 0 but \(z^2\) belongs to the closure of the ideal of \(R(X)\)-functions vanishing in a neighborhood of the origin. Another Swiss cheese \(X\) is constructed with a property that \(R(X)\) contains a prime ideal \(P\) whose closure \(\overline P\) is not prime.

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
41A20 Approximation by rational functions
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