Submitted manuscript, 541 KB, PDF document
Accepted author manuscript, 551 KB, PDF document
Research output: Contribution to Journal/Magazine › Journal article › peer-review
| <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 |
| Publication Status | 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 FE(); (iii) the answer to Question (II) is positive for many, but not all, Banach spaces.