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Commutators on \(\ell _1\). (English) Zbl 1170.47022

The classification of commutators, that is, of elements of the form \(AB-BA\) in a given Banach algebra, has a long history. Restricting ourselves to the case of \(B(X)\), the algebra of all operators on the Banach space \(X\), the first breakthrough was made by A.Brown and C.Pearcy [Ann.Math.(2) 82, 112–127 (1965; Zbl 0131.12302)] who proved that for a Hilbert space \(H\), the only operators on \(H\) that are not commutators are those of the form \(\lambda I + K\), with \(K\) compact. C.Apostol [Rev.Roum.Math.Pures Appl.17, 1513–1534 (1972; Zbl 0247.47030)] extended the Brown–Pearcy result to \(X = \ell_p\) with \(1 < p < \infty\).
The main result of the present paper is that the commutators on \(\ell_1\) are the operators not of the form \(\lambda I + K\), with \(\lambda \neq 0\) and \(K\) compact. Other results are also proved.

MSC:

47B47 Commutators, derivations, elementary operators, etc.
46B25 Classical Banach spaces in the general theory
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References:

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