Discontinuous Homomorphisms from Non‐Commutative Banach Algebras

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 Φ A of A is infinite [ 3 , Theorem 3] and also whenever there is a non‐maximal, prime ideal P in A such that | A / P | = 2 ℵ 0[ 4 , 8 ] . (It is an open question whether or not every infinite‐dimensional, commutative Banach algebra A satisfies this latter condition.) 1991 Mathematics Subject Classification 46H40.