Generators of maximal left ideals in Banach algebras

H. G. Dales, and W. Żelazko. "Generators of maximal left ideals in Banach algebras." Studia Mathematica 212.2 (2012): 173-193. <http://eudml.org/doc/285554>.

@article{H2012,
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. },
author = {H. G. Dales, W. Żelazko},
journal = {Studia Mathematica},
keywords = {maximal left ideal; finite-dimensional Banach algebras; finitely generated ideals; countably generated ideals; Shilov boundary},
language = {eng},
number = {2},
pages = {173-193},
title = {Generators of maximal left ideals in Banach algebras},
url = {http://eudml.org/doc/285554},
volume = {212},
year = {2012},
}

TY - JOUR
AU - H. G. Dales
AU - W. Żelazko
TI - Generators of maximal left ideals in Banach algebras
JO - Studia Mathematica
PY - 2012
VL - 212
IS - 2
SP - 173
EP - 193
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.
LA - eng
KW - maximal left ideal; finite-dimensional Banach algebras; finitely generated ideals; countably generated ideals; Shilov boundary
UR - http://eudml.org/doc/285554
ER -