This is an announcement for the paper "Maximal left ideals of the Banach algebra of bounded operators on a Banach space" by H. G. Dales, Tomasz Kania, Tomasz Kochanek, Piotr Koszmider, and Niels Jakob Laustsen.
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 space $E$: (I) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (II) 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 (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 $\mathscr{B}(E)$ contains $\mathscr{F}(E)$; (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.
Archive classification: math.FA math.OA
Mathematics Subject Classification: Primary 47L10, 46H10, Secondary 47L20
Submitted from: t.kania@lancaster.ac.uk
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1208.4762
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