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Published online by Cambridge University Press: 20 November 2018
We show that ${{L}^{\infty }}\left( \mu \right)$, in its capacity as multiplication operators on
${{L}^{p}}\left( \mu \right)$, is minimal as a
$p$-operator space for a decomposable measure
$\mu $. We conclude that
${{L}^{1}}\left( \mu \right)$ has a certain maximal type
$p$-operator space structure that facilitates computations with
${{L}^{1}}\left( \mu \right)$ and the projective tensor product.