  Given operator spaces $V$ and $W$, let $\widetilde{W}$ denote the opposite
operator space structure on the same underlying Banach space. Although the
identity map $W\to \widetilde{W}$ is in general not completely bounded, we show
that the identity map on $V\otimes W$ extends to a contractive linear map
$V\widehat{\otimes} W \to V\otimes_{\min} \widetilde{W}$, where
$\widehat{\otimes}$ and $\otimes_{\rm min}$ denote the projective and injective
tensor products of operator spaces. In future work, this will be applied to
construct antisymmetric $2$-cocycles on certain Fourier algebras.
