Maximal left ideals of the Banach algebra of bounded operators on a Banach space (Q2859337)

The first author and \textit{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 \(E\)).NEWLINENEWLINEThe work is structured around two questions:NEWLINENEWLINE(I) Does \(\mathcal B(E)\) always contain a maximal left ideal which is not finitely generated?NEWLINENEWLINE(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 \(x\not= 0\))?NEWLINENEWLINERegarding 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.NEWLINENEWLINETurning 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.NEWLINENEWLINEFacing 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 \textit{S. A. Argyros} and \textit{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).NEWLINENEWLINESome interesting open questions are collected at the final section.