Discontinuous homomorphisms from non-commutative Banach algebras

Dales, H.G. and Runde, Volker (1997) Discontinuous homomorphisms from non-commutative Banach algebras. Bulletin of the London Mathematical Society, 29 (4). pp. 475-479. ISSN 0024-6093

Abstract

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 ∣A/P∣=2ℵ0 [4, 8]. (It is an open question whether or not every infinite-dimensional, commutative Banach algebra A satisfies this latter condition.)