my forty years on his shoulders - Department of Mathematics

5concerning automatic continuity. In one of its more concrete special forms, itasserts that*) every homomorphism from the Banach algebra c 0 of infinite sequencesof reals converging to 0 (under the sup norm) to any separable Banach algebra, iscontinuous.Now *) was refuted using the continuum hypothesis (due independentlyto H.G. Dales and J. Esterle), and later shown to be not refutable without thecontinuum hypothesis; i.e., not refutable in the usual ZFC axioms (due to R.Solovay). See (Dales 2001) for the refutation, and (Dales, Woodin 1987) for theconsistency (non refutability) result.It is, of course, much easier for mathematicians to recognize the overtly settheoretic character after they learn that there are set theoretic difficulties. Bytaking the negation,**) there exists a discontinuous homomorphism from the Banach algebrac 0 of infinite sequences of reals converging to 0 (under the sup norm) to someseparable Banach algebra.It is clear that one is asking about the existence of an object that was wellknown, even at the time, to necessarily have rather pathological properties. Thisis the case even for discontinuous group homomorphisms from R into R (whichcan be shown to exist without the continuum hypothesis). For instance, it is wellknown that there are no discontinuous group homomorphisms from R into Rthat are Borel measurable.