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Abstract

Let $E$ be a Riesz space. By defining the spaces $L_{E}^{1}$ and $L_{E}^{\infty }$ of $E$, we prove that the center $Z(L_{E}^{1})$ of $L_{E}^{1}$ is $L_{E}^{\infty }$ and show that the injectivity of the Arens homomorphism $m:Z(E)^{\prime \prime }\rightarrow Z(E^{\sim })$ is equivalent to the equality $L_{E}^{1}=Z(E)^{\prime }$. Finally, we also give some representation of an order continuous Banach lattice $E$ with a weak unit and of the order dual $E^{\sim }$ of $E$ in $L_{E}^{1}$ which are different from the representations appearing in the literature.