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DISCONTINUOUS HOMOMORPHISMS FROM NON-COMMUTATIVE BANACH ALGEBRAS

Published online by Cambridge University Press:  01 July 1997

H. G. DALES  and
V. RUNDE
H. G. DALES
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT
V. RUNDE
Affiliation:
Fachbereich 9 Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
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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 [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.)

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 29 , Issue 4 , July 1997 , pp. 475 - 479
Copyright
© The London Mathematical Society 1997

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