Kaplansky's and Michael's problems‎: ‎a survey

Abstract

‎I‎. ‎Kaplansky showed in 1947 that every submultiplicative norm $\Vert‎ . ‎\Vert$ on the algebra ${\mathcal C}(K)$ of complex--valued‎ ‎functions on an infinite compact space $K$ satisfies $\Vert f \Vert‎ ‎\ge \Vert f \Vert _K$ for every $f \in {\mathcal C}(K),$ where‎ ‎$\Vert f \Vert _K=max_{t \in K} \vert f(t)\vert $ denotes the‎ ‎standard norm on ${\mathcal C}(K).$ He asked whether all‎ ‎submultiplicative norms $\Vert‎ . ‎\Vert$ were in fact equivalent to‎ ‎the standard norm (which is obviously true for finite compact‎ ‎spaces)‎, ‎or equivalently‎, ‎whether all homomorphisms from ${\mathcal‎ ‎C}(K)$ into a Banach algebra were continuous‎. ‎This problem turned‎ ‎out to be undecidable in ZFC‎, ‎and we will discuss here some recent‎ ‎progress due to Pham and open questions concerning the structure of‎ ‎the set of nonmaximal prime ideals of ${\mathcal C}(K)$ which are‎ ‎closed with respect to a discontinuous submultiplicative norm on‎ ‎${\mathcal C}(K)$ when the continuum hypothesis is assumed‎. ‎We will‎ ‎also discuss the existence of discontinuous characters on Fr\'echet‎ ‎algebras (Michael's problem)‎, ‎a long standing problem which remains‎ ‎unsolved‎. ‎The Mittag--Leffler theorem on inverse limits of complete‎ ‎metric spaces plays an essential role in the literature concerning‎ ‎both problems‎.