When studying Fourier algebras of locally compact groups, it is commonly accepted that we need to use the projective tensor product of operator spaces. On the other hand, the study of derivations on Fourier algebras has revealed a potentially rich area for investigation, but such derivations can never be completely bounded, and hence there would seem to be no reason why they should interact well with operator space tensor products.

In this talk I will explain more about these two themes, and why they presented an obstacle until recently when trying to construct non-trivial 2-cocycles on Fourier algebras. I will then outline how the obstacle can be overcome by making use of extra structure for certain derivations, together with a "twisted inclusion" result for operator space tensor products.