  We show that $L^\infty(\mu)$, in its capacity as multiplication operators on
$L^p(\mu)$, is minimal as a $p$-operator space for a decomposable measure
$\mu$. We conclude that $L^1(\mu)$ has a certain maximal type $p$-operator
space structure which facilitates computations with $L^1(\mu)$ and the
projective tensor product.
