Final published version, 348 KB, PDF-document

- 10.4064/sm212-2-5
Final published version

Research output: Contribution to journal › Journal article

Published

<mark>Journal publication date</mark> | 2012 |
---|---|

<mark>Journal</mark> | Studia Mathematica |

Issue number | 2 |

Volume | 212 |

Number of pages | 21 |

Pages (from-to) | 173-193 |

<mark>State</mark> | Published |

<mark>Original language</mark> | English |

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.

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.