Minimal and maximal $p$-operator space structures

We show that $L^{\mathrm{\infty }}(\mu )$, 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.