In the 1970s, a question of Kaplansky about discontinuous homomorphisms from
certain commutative Banach algebras was resolved. Let A be the
commutative C*-algebra C(Ω), where Ω is an
infinite compact space. Then, if the continuum hypothesis
(CH) be assumed, there is a discontinuous homomorphism from
C(Ω) into a Banach
algebra [2, 7]. In fact, let A
be a commutative Banach algebra. Then (with (CH)) there
is a discontinuous homomorphism from A into a Banach algebra whenever the
character space ΦA of A is infinite [3,
Theorem 3] and also
whenever there is a non-maximal, prime ideal P in A such that
[mid ]A/P[mid ]=2ℵ0
[4, 8]. (It is an open question
whether or not every infinite-dimensional, commutative Banach algebra
A satisfies this latter condition.)