Rights statement: 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 https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms.12290 This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.
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Final published version
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| <mark>Journal publication date</mark> | 29/02/2020 |
|---|---|
| <mark>Journal</mark> | Bulletin of the London Mathematical Society |
| Issue number | 1 |
| Volume | 52 |
| Number of pages | 15 |
| Pages (from-to) | 1-15 |
| Publication Status | Published |
| Early online date | 1/12/19 |
| <mark>Original language</mark> | English |
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?