Discontinuous Homomorphisms from C ( X

The conjecture that every algebra norm || • || on C(X) is equivalent to the uniform norm arises naturally from a theorem of Kaplansky in 1949 that necessarily 11/11 > \f\x ( ƒ e C(X)): see [9, 10.1]. The seminal paper on the automatic continuity of homomorphisms from C(X) is the 1960 paper of Bade and Curtis [2] in which, for example, it is proved that there is a discontinuous homomorphism from C(X) if and only if there is a radical homomorphism, a nonzero homomorphism from a maximal ideal of C(X) into a commutative radical Banach algebra. In 1967, it was proved by Johnson that every homomorphism from certain noncommutative C*-algebras is continuous: see [9, 12.4]. Sinclair proved recently that the existence of a discontinuous homomorphism is equivalent to the existence of an algebra norm on C(X)/I for some nonmaximal prime ideal I of C(X) [9, 11.7], and this was proved independently in [4]. It follows from the work of each of the present authors that such a norm exists provided \C(X)/I\ = Kj. Assuming CH, such an ideal exists for each X, and every nonmaximal prime ideal has this property if X is separable, but, if X is not separable, there may exist a prime ideal / such that C(X)/J is not normable [4].