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arXiv · 1208.4762

Maximal left ideals of the Banach algebra of bounded operators on a Banach space

H. G. Dales·Tomasz Kania·Tomasz Kochanek·Piotr Koszmider·Niels Jakob Laustsen
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Abstract

We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (Q1) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (Q2) has a positive answer if and only if no finitely-generated, maximal left ideal of $\mathscr{B}(E)$ contains $\mathscr{F}(E)$; (iii) the answer to Question (Q2) is positive for many, but not all, Banach spaces.

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H. G. Dales, Tomasz Kania, Tomasz Kochanek, Piotr Koszmider, Niels Jakob Laustsen. 2013-09-11. Maximal left ideals of the Banach algebra of bounded operators on a Banach space. https://doi.org/10.4064/sm218-3-3

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