Geodesic
Generators of maximal left ideals in Banach algebras
H. G. Dales 1   ; W. Żelazko 2
1 Department of Mathematics and Statistics Fylde College University of Lancaster Lancaster LA1 4YF, United Kingdom
2 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland
Studia Mathematica, Tome 212 (2012) no. 2, pp. 173-193 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

Voir la notice de l'article

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 $\mathbb C$ 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.
DOI : 10.4064/sm212-2-5
Mots-clés : grauert remmert proved commutative complex noetherian banach algebra necessarily finite dimensional precisely proved commutative complex banach algebra has finite dimension mathbb whenever closed ideals algebra algebraically finitely generated sinclair tullo obtained non commutative version result ferreira tomassini improved result grauert remmert showing statement replaces closed ideals maximal ideals shilov boundary shorter proof latter result together extensions related examples study following conjecture suppose maximal ideals unital banach algebra finitely generated finite dimensional

H. G. Dales  1   ; W. Żelazko  2

1 Department of Mathematics and Statistics Fylde College University of Lancaster Lancaster LA1 4YF, United Kingdom
2 Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland 
@article{10_4064_sm212_2_5,
     author = {H. G. Dales and W. \.Zelazko},
     title = {Generators of maximal left ideals in  {Banach} algebras},
     journal = {Studia Mathematica},
     pages = {173--193},
     year = {2012},
     volume = {212},
     number = {2},
     doi = {10.4064/sm212-2-5},
     language = {de},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/sm212-2-5/}
}
TY  - JOUR
AU  - H. G. Dales
AU  - W. Żelazko
TI  - Generators of maximal left ideals in  Banach algebras
JO  - Studia Mathematica
PY  - 2012
SP  - 173
EP  - 193
VL  - 212
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4064/sm212-2-5/
DO  - 10.4064/sm212-2-5
LA  - de
ID  - 10_4064_sm212_2_5
ER  - 
%0 Journal Article
%A H. G. Dales
%A W. Żelazko
%T Generators of maximal left ideals in  Banach algebras
%J Studia Mathematica
%D 2012
%P 173-193
%V 212
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4064/sm212-2-5/
%R 10.4064/sm212-2-5
%G de
%F 10_4064_sm212_2_5
H. G. Dales; W. Żelazko. Generators of maximal left ideals in  Banach algebras. Studia Mathematica, Tome 212 (2012) no. 2, pp. 173-193. doi: 10.4064/sm212-2-5

Cité par Sources :