The Bishop–Phelps–Bollobás theorem for operators on L1(μ)L1(μ)

In this paper we show that the Bishop–Phelps–Bollobás theorem holds for L(L1(μ),L1(ν))L(L1(μ),L1(ν)) for all measures μ and ν   and also holds for L(L1(μ),L∞(ν))L(L1(μ),L∞(ν)) for every arbitrary measure μ and every localizable measure ν  . Finally, we show that the Bishop–Phelps–Bollobás theorem holds for two classes of bounded linear operators from a real L1(μ)L1(μ) into a real C(K)C(K) if μ is a finite measure and K is a compact Hausdorff space. In particular, one of the classes includes all Bochner representable operators and all weakly compact operators.