Minimal and maximal $p$-operator space structures

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

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