Maximal chains of closed prime ideals for discontinuous algebra norms on \(\mathcal C (K)\) (Q2861058)

\textit{I. Kaplansky} asked in [Duke Math. J. 16, 399--418 (1949; Zbl 0033.18701)] whether or not every algebra norm on the algebra \(C(K)\) of continuous functions on an infinite compact space \(K\) is continuous with respect to the natural uniform norm on \(C(K)\); this is equivalent to whether or not every homomorphism from \(C(K)\) into a Banach algebra is continuous. This famous problem has been one of the major driving forces of the development of Banach algebra theory, in which the author is one of the main contributors. Kaplansky's problem was eventually solved in the negative, independently by the author and H. G. Dales in the late 1970s ([\textit{H. G. Dales}, Am. J. Math. 101, 647--734 (1979; Zbl 0417.46054)], [\textit{J. Esterle}, Proc. Lond. Math. Soc., III. Ser. 36, 59--85 (1978; Zbl 0411.46039)]); they gave not only a counter-example, but actually proved a general result that, assuming the continuum hypothesis, every (non-maximal) prime ideal in \(C(K)\) is the continuity ideal of a discontinuous homomorphism into the unitization of some radical Banach algebra. On the other hand, R.\,M.\thinspace Solovay and W.\,H.\thinspace Woodin constructed a model of set theory in which all homomorphisms from \(C(K)\) are continuous (see [\textit{H. G. Dales} and \textit{W. H. Woodin}, An introduction to independence for analysts. Cambridge University Press (1987; Zbl 0629.03030)] for details). Note also that there are models of set theory in which \(2^{\aleph_0}=\aleph_2\), but in which discontinuous homomorphisms from \(C(K)\) exist ([\textit{R. Frankiewicz} and \textit{P. Zbierski}, Hausdorff gaps and limits. Amsterdam: North-Holland (1994; Zbl 0821.54001)] and [\textit{W. H. Woodin}, J. Lond. Math. Soc., II. Ser. 48, No. 2, 299--315 (1993; Zbl 0804.46063)]).NEWLINENEWLINEThus it is of great interest to completely determine the structure of (discontinuous) homomorphisms from \(C(K)\) or, what amounts to the same thing, of (discontinuous) algebra norms on \(C(K)\). Starting with \textit{W. G. Bade} and \textit{P. C. Curtis jun.} [Am. J. Math. 82, 589--608 (1960; Zbl 0093.12503)], many partial results have been obtained in this direction, and the paper under review is an important contribution towards this ultimate aim. The main result of this paper gives the construction of a discontinuous algebra norm whose collection of closed prime ideals is any given, not necessarily countable, well-ordered chain (satisfying a minor assumption).