Injective dual Banach Spaces and Operator Ideals

We give characterizations of Banach spaces with -injective biduals in terms ofoperators with an integral representation. These results may be thought of as acertain refinement of Lindenstrauss’ extension theorems from his 1964 memoir.We also obtain dual results characterizing Banach spaces with -injective dualsin terms of factorization of operators through or . AQ1 Keywords -Injective spaces Extension property Operators with an integral representation -spaces -spaces Factorization through or Mathematics Subject Classification Primary 47B10 Secondary 46B03 46B28 To the memory of our beloved and admired Professor Eve Oja. Raffaella Cilia was supported in part by G.N.A.M.P.A. (Italy). Raffaella Cilia,Joaquín M. Gutiérrez were supported in part by Secretaría de Estado deInvestigación, Desarrollo e Innovación, PGC2018-097286-B-I00 (Spain). 1. Introduction Given Banach spaces X and Y , we denote by the Banach space of all(linear bounded) operators from X into Y endowed with the operator norm. We use (respectively, ) for the subspace of compact (respectively,weakly compact) operators from X into Y . It is well known that, given a compact Hausdorff space K , every operator admits an integral representation [ 10 , Theorem VI.2.1]. λ λ L1(μ) ℓ1