A twisted inclusion between tensor products of operator spaces

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.