Dosev, D.; Johnson, W. B. Commutators on \(\ell_\infty\). (English) Zbl 1190.47036 Bull. Lond. Math. Soc. 42, No. 1, 155-169 (2010). The commutator of two elements \(A\) and \(B\) in a Banach algebra \({\mathcal A}\) is given by \([A,B]=AB-BA\). A natural problem is to classify the commutators of a given algebra. Of particular interest is studying this question for the Banach algebra of linear operators on the sequence space \(\ell^p\) with \(1\leq p\leq\infty\). In [Ann.Math.(2) 82, 112–127 (1965; Zbl 0131.12302)], A.Brown and C.Pearcy demonstrated that when \(p=2\), the set of operators which are not commutators are of the form \(\lambda I+K\), where \(\lambda\neq 0\) and \(K\) is a compact operator. This was later extended to \(1<p<\infty\) by C.Apostol [Rev.Roum.Math.Pures Appl.17, 1513–1534 (1972; Zbl 0247.47030)]. Subsequently, C.Apostol dealt with the classification of commutators on \(c_0\) [Rev.Roum.Math.Pures Appl.18, 1025–1032 (1973; Zbl 0261.47019)]. The case of \(p=1\) was settled by the first author [J. Funct.Anal 256, No.11, 3490–3509 (2009; Zbl 1170.47022)] with the same classification of non-commutators.This well-written paper undertakes the challenge of determining the characterization of the commutators when \(p=\infty\). In particular, the authors demonstrate that the linear operators on \(\ell^\infty\) that are commutators are not of the form \(\lambda I +S\) with \(\lambda\neq 0\) and \(S\) a strictly singular operator.The rough idea of proof is the following. First, the authors point out that the classification of the commutators on \(\ell^p\) when \(1\leq p<\infty\) leads to the following conjecture.Conjecture. Let \({\mathcal X}\) be a Banach space such that \({\mathcal X}\simeq(\sum{\mathcal X})_p\), \(1\leq p\leq\infty\) or \(p=0\) (the authors say that the space \({\mathcal X}\) then admits a Pełczyński decomposition). Assume that \({\mathcal L}({\mathcal X})\) has a largest ideal \({\mathcal M}\). Then every non-commutator on \({\mathcal X}\) has the form \(\lambda I +K\), where \(K\in{\mathcal M}\) and \(\lambda\neq 0\). In the case of \(\ell^\infty\), the authors use for \({\mathcal M}\) the collection of all strictly singular operators. Then they demonstrate that every operator on \(\ell^\infty\) which is strictly singular is a commutator. They also show that, if \(T\in{\mathcal L}(\ell^\infty)\) is not of the form \(\lambda I +S\), where \(S\) is strictly singular and \(\lambda\neq 0\), then \(T\) is a commutator. Reviewer: Brett Wick (Atlanta) Cited in 16 Documents MSC: 47B47 Commutators, derivations, elementary operators, etc. 47L20 Operator ideals Citations:Zbl 0131.12302; Zbl 0247.47030; Zbl 0261.47019; Zbl 1170.47022 PDFBibTeX XMLCite \textit{D. Dosev} and \textit{W. B. Johnson}, Bull. Lond. Math. Soc. 42, No. 1, 155--169 (2010; Zbl 1190.47036) Full Text: DOI arXiv