## 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

### Citations:

Zbl 1269.46028; Zbl 1223.46007
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