Laustsen, Niels J. (2002) *Maximal ideals in the algebra of operators on certain Banach spaces.* Proceedings of the Edinburgh Mathematical Society, 45 (3). pp. 523-546. ISSN 0013-0915

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## Abstract

For a Banach space $\mathfrak{X}$, let $\mathcal{B}(\mathfrak{X})$ denote the Banach algebra of all continuous linear operators on $\mathfrak{X}$. First, we study the lattice of closed ideals in $\mathcal{B}(\mathfrak{J}_p)$, where $1 < p < \infty$ and $\mathfrak{J}_p$ is the $p$th James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{J}_p)$. Applications of this result include the following. (i) The Brown–McCoy radical of $\mathcal{B}(\mathfrak{X})$, which by definition is the intersection of all maximal ideals in $\mathcal{B}(\mathfrak{X})$, cannot be turned into an operator ideal. This implies that there is no ‘Brown–McCoy’ analogue of Pietsch’s construction of the operator ideal of inessential operators from the Jacobson radical of $\mathcal{B}(\mathfrak{X})/\mathcal{A}(\mathfrak{X})$. (ii) For each natural number $n$ and each $n$-tuple $(m_1,\dots,m_n)$ in $\{k^2\mid k\in\mathbb{N}\}\cup\{\infty\}$, there is a Banach space $\mathfrak{X}$ such that $\mathcal{B}(\mathfrak{X})$ has exactly $n$ maximal ideals, and these maximal ideals have codimensions $m_1,\dots,m_n$ in $\mathcal{B}(\mathfrak{X})$, respectively; the Banach space $\mathfrak{X}$ is a finite direct sum of James spaces and $\ell_p$-spaces. Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces $\mathfrak{X}$ such that all the maximal ideals in $\mathcal{B}(\mathfrak{X})$ can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{X})$ for each hereditarily indecomposable Banach space $\mathfrak{X}$, and we prove that there are $2^{2^{\aleph_0}}$ distinct maximal ideals in $\mathcal{B}(\mathfrak{G})$, where $\mathfrak{G}$ is the Banach space constructed by Gowers to solve Banach’s hyperplane problem.

Item Type: | Article |
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Journal or Publication Title: | Proceedings of the Edinburgh Mathematical Society |

Additional Information: | The final, definitive version of this article has been published in the Journal, Proceedings of the Edinburgh Mathematical Society, 45 (3), pp 523-546 2002, © 2002 Cambridge University Press. RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics |

Subjects: | Q Science > QA Mathematics |

Departments: | Faculty of Science and Technology > Mathematics and Statistics |

ID Code: | 2379 |

Deposited By: | ep_importer |

Deposited On: | 01 Apr 2008 11:14 |

Refereed?: | Yes |

Published?: | Published |

Last Modified: | 18 Jan 2017 01:28 |

Identification Number: | |

URI: | http://eprints.lancs.ac.uk/id/eprint/2379 |

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