Weak∗ exactness for dual operator spaces

We introduce a new tensor product and study the weak∗ condition C′, which is also called weak∗ exactness, for dual operator spaces. Our definition of weak∗ condition C′ is equivalent to Kirchberg's notion of weak exactness in the case of von Neumann algebras. We also study the connection of weak∗ exact W∗-TROs with their linking von Neumann algebras and study the structure of exact (respectively, nuclear) W∗-TROs.