  Let $\mathcal{M}$ be a von Neumann algebra, and let $0<p,q\le\infty$. Then
the space $\Hom_\mathcal{M}(L^p(\mathcal{M}),L^q(\mathcal{M}))$ of all right
$\mathcal{M}$-module homomorphisms from $L^p(\mathcal{M})$ to
$L^q(\mathcal{M})$ is a reflexive subspace of the space of all continuous
linear maps from $L^p(\mathcal{M})$ to $L^q(\mathcal{M})$. Further, the space
$\Hom_\mathcal{M}(L^p(\mathcal{M}),L^q(\mathcal{M}))$ is hyperreflexive in each
of the following cases: (i) $1\le q<p\le\infty$; (ii) $1\le p,q\le\infty$ and
$\mathcal{M}$ is injective, in which case the hyperreflexivity constant is at
most $8$.
