Geodesic
A Generalized Fredholm Theory for Certain Maps in the Regular Representations of an Algebra
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 495-504

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Given an algebra A, the elements of A induce linear operators on A by left and right multiplication. Various authors have studied Banach algebras A with the property that some or all of these multiplication maps are completely continuous operators on A ; see (1-5). In (3)1. Kaplansky defined an element u of a Banach algebra A to be completely continuous if the maps a ⟶ ua and a ⟶ au, a ∊ A, are completely continuous linear operators. 
Barnes, Bruce Alan. A Generalized Fredholm Theory for Certain Maps in the Regular Representations of an Algebra. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 495-504. doi: 10.4153/CJM-1968-048-2
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