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Generators of maximal left ideals in Banach algebras

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Generators of maximal left ideals in Banach algebras. / Dales, H.G.; Zelazko, W.
In: Studia Mathematica, Vol. 212, No. 2, 2012, p. 173-193.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Dales, HG & Zelazko, W 2012, 'Generators of maximal left ideals in Banach algebras', Studia Mathematica, vol. 212, no. 2, pp. 173-193. https://doi.org/10.4064/sm212-2-5

APA

Dales, H. G., & Zelazko, W. (2012). Generators of maximal left ideals in Banach algebras. Studia Mathematica, 212(2), 173-193. https://doi.org/10.4064/sm212-2-5

Vancouver

Dales HG, Zelazko W. Generators of maximal left ideals in Banach algebras. Studia Mathematica. 2012;212(2):173-193. doi: 10.4064/sm212-2-5

Author

Dales, H.G. ; Zelazko, W. / Generators of maximal left ideals in Banach algebras. In: Studia Mathematica. 2012 ; Vol. 212, No. 2. pp. 173-193.

Bibtex

@article{a33ccee193024b909b1fa3fbd00e09bf,
title = "Generators of maximal left ideals in Banach algebras",
abstract = "In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over C whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Shilov boundary of A'. We give a shorter proof of this latter result, together with some extensions and related examples.We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra A are finitely generated. Then A is finite-dimensional. ",
keywords = "Banach algebra, maximal left ideal",
author = "H.G. Dales and W. Zelazko",
year = "2012",
doi = "10.4064/sm212-2-5",
language = "English",
volume = "212",
pages = "173--193",
journal = "Studia Mathematica",
issn = "0039-3223",
publisher = "Instytut Matematyczny",
number = "2",

}

RIS

TY - JOUR

T1 - Generators of maximal left ideals in Banach algebras

AU - Dales, H.G.

AU - Zelazko, W.

PY - 2012

Y1 - 2012

N2 - In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over C whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Shilov boundary of A'. We give a shorter proof of this latter result, together with some extensions and related examples.We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra A are finitely generated. Then A is finite-dimensional.

AB - In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over C whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Shilov boundary of A'. We give a shorter proof of this latter result, together with some extensions and related examples.We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra A are finitely generated. Then A is finite-dimensional.

KW - Banach algebra

KW - maximal left ideal

U2 - 10.4064/sm212-2-5

DO - 10.4064/sm212-2-5

M3 - Journal article

VL - 212

SP - 173

EP - 193

JO - Studia Mathematica

JF - Studia Mathematica

SN - 0039-3223

IS - 2

ER -