The Bishop-Phelps-Bollobás theorem for operators on $L_1(μ)$

In this paper we show that the Bishop-Phelps-Bollobás theorem holds for $\mathcal{L}(L_1(μ), L_1(ν))$ for all measures $μ$ and $ν$ and also holds for $\mathcal{L}(L_1(μ),L_\infty(ν))$ 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 $L_1(μ)$ into a real $C(K)$ 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.

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