On the duals of Lebesgue-Bochner 𝐿^{𝑝} spaces

Abstract

Let ( X , A , μ ) (X,\mathcal {A},\mu ) be an arbitrary positive measure space. We prove that there exist an extremally disconnected (locally) compact Hausdorff space Y Y and a perfect (regular) Borel measure ν \nu on Y Y such that L p ( μ , E ) ≃ L p ( ν , E ) {L^p}(\mu ,\textrm {E}) \simeq {L^p}(\nu ,E) for all 1 ≤ p > ∞ 1 \leq p > \infty and any Banach space E E . If E ∗ {E^*} is separable, then L p ( μ , E ) ∗ ≃ L q ( μ , E ∗ ) {L^p}(\mu ,\textrm {E})* \simeq {L^q}(\mu ,{\textrm {E}^*}) for all 1 > p > ∞ , 1 p + 1 q = 1 1 > p > \infty ,\;\frac {1}{p} + \frac {1}{q} = 1 , and L 1 ( μ , E ) ∗ ≃ L ∞ ( ν , E ∗ ) ≃ C ( β Y , E ∗ ∗ ) {L^1}(\mu ,\textrm {E})* \simeq {L^\infty }(\nu ,{\textrm {E}^*}) \simeq C(\beta Y,\textrm {E}_*^*) , where E ∗ ∗ E_*^* denotes E ∗ {E^*} endowed with the weak* topology. In particular L 1 ( μ ) ∗ ≃ L ∞ ( ν ) {L^1}{(\mu )^*} \simeq {L^\infty }(\nu ) .