Maximal left ideals in Banach algebras

Cabrera Garcia, M. and Dales, H. G. and Rodríguez-Palacios, A. (2020) Maximal left ideals in Banach algebras. Bulletin of the London Mathematical Society, 52 (1). pp. 1-15. ISSN 0024-6093

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Let A be a Banach algebra. Then frequently each maximal left ideal in A is closed, but there are easy examples that show that a maximal left ideal can be dense and of codimension 1 in A. It has been conjectured that these are the only two possibilities: each maximal left ideal in a Banach algebra A is either closed or of codimension 1 (or both). We shall show that this is the case for many Banach algebras that satisfy some extra condition, but we shall also show that the conjecture is not always true by constructing, for each n is an element of N, examples of Banach algebras that have a dense maximal left ideal of codimension n. In particular, we shall exhibit a semi-simple Banach algebra with this property. We shall show that the questions concerning maximal left ideals in a Banach algebra A that we are considering are related to automatic continuity questions: When are A-module homomorphisms from A into simple Banach left A-modules automatically continuous?

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Journal Article
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Bulletin of the London Mathematical Society
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This is the peer reviewed version of the following article: García, M.C., Dales, H.G. and Palacios, Á.R. (2020), Maximal left ideals in Banach algebras. Bull. London Math. Soc., 52: 1-15. doi:10.1112/blms.12290 which has been published in final form at This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.
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08 Oct 2019 09:20
Last Modified:
07 May 2021 04:31