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

Research output: Contribution to journalJournal article

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Journal publication date2013
JournalStudia Mathematica
Journal number3
Volume218
Number of pages42
Pages245-286
Original languageEnglish

Abstract

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 FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.