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Maximal left ideals of the Banach algebra of bounded operators on a Banach space. (English) Zbl 1308.47088

The first author and W. Ẓelazko [ibid. 212, No. 2, 173–193 (2012; Zbl 1269.46028)] have conjectured that any infinite dimensional unital Banach algebra \(\mathcal A\) should contain a maximal left ideal which is not finitely generated. In the beautiful paper under review, the authors deeply study this conjecture for the relevant particular case when \(\mathcal A=\mathcal B(E)\) (the algebra of bounded linear operators on an infinite dimensional Banach space \(E\)).
The work is structured around two questions:
(I) Does \(\mathcal B(E)\) always contain a maximal left ideal which is not finitely generated?
(II) Is every finitely-generated, maximal left ideal of \(\mathcal B(E)\) necessarily fixed (that is, of the form \(\{T\in \mathcal B(E): Tx=0\}\), for certain \(x\not= 0\))?
Regarding the first question, they prove that the answer is affirmative when \(E\) is a separable Banach space with unconditional Schauder decomposition. Moreover, in such case, they show that the cardinality of the set of non finitely-generated maximal left ideals of \(\mathcal B(E)\) is \(2^{\mathfrak{c}}\) (with \(\mathfrak{c}=2^{\aleph_0}\)). Some other examples or conditions for a positive answer to question (I) are given. Its general validity remains open.
Turning to the second question, they present in Theorem 1.1 the following dichotomy for a maximal left ideal \(\mathcal L\) of \(\mathcal B(E)\): either \(\mathcal L\) is fixed or it contains all finite-rank operators. Through this result (and a lot of specific work developed in Sections 5 and 6) they are able to find many examples of classical (and also “exotic”) Banach spaces \(E\) for which the answer to question (II) is positive. That is the case for instance when \(E\) is a Hilbert space, \(E=c_0\), \(E\) is an injective Banach space or \(E\) a Banach space with few operators.
Facing this bunch of positive examples, they build in Section 7 an example of a Banach space \(E\) for which the answer to question (II) is negative. They begin with the space \(X_{AH}\) constructed by S. A. Argyros and R. G. Haydon [Acta Math. 206, No. 1, 1–54 (2011; Zbl 1223.46007)] and consider \(E=X_{AH}\oplus_\infty\ell_\infty\). They prove also that this last space fails to provide an example of a negative answer to question (I).
Some interesting open questions are collected at the final section.

MSC:

47L10 Algebras of operators on Banach spaces and other topological linear spaces
46H10 Ideals and subalgebras
47L20 Operator ideals
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