**Maximal left ideals of the Banach algebra of bo...**541 KB, PDF-document

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Research output: Contribution to journal › Journal article

Published

<mark>Journal publication date</mark> | 2013 |
---|---|

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

Issue number | 3 |

Volume | 218 |

Number of pages | 42 |

Pages (from-to) | 245-286 |

<mark>State</mark> | Published |

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

We address the following two questions regarding the maximal left ideals of the Banach algebra *B*(*E*) of bounded operators acting on an infinite-dimensional Banach space *E*:

(I) Does *B*(*E*) always contain a maximal left ideal which is not finitely generated?

(II) Is every finitely-generated, maximal left ideal of *B*(*E*) necessarily of the form {*T* in *B*(*E*) : *Tx *= 0} for some non-zero *x* in *E*?

Since the two-sided ideal *F*(*E*) of finite-rank operators is not contained in any of the maximal left ideals described in (II), a positive answer to the second question would imply a positive answer to the first.

Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of *B*(*E*) contains *F*E* (*); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.