Operator space projective tensor product: Embedding into second dual and ideal structure

We prove that for operator spaces $V$ and $W$, the operator space $V^{\ast \ast }{\otimes }_h{W}^{\ast \ast }$ can be completely isometrically embedded into $(V{\otimes }_{h}W{)}^{\ast \ast }$, ${\otimes }_h$ being the Haagerup tensor product. It is also shown that, for exact operator spaces $V$ and $W$, a jointly completely bounded bilinear form on $V\times W$ can be extended uniquely to a separately $w^{\ast }$-continuous jointly completely bounded bilinear form on $ V^{}\times W^{}$\mathrm{.}Thispavesthewaytoobtainacanonicalembeddingof$V^{}\hat{\otimes} W^{}$into$(V\hat{\otimes} W)^{**}$ with a continuous inverse, where $\widehat{\otimes }$ is the operator space projective tensor product. Further, for $C^{\ast }$-algebras $A$ and $B$, we study the (closed) ideal structure of $A\widehat{\otimes }B$, which, in particular, determines the lattice of closed ideals of $B(H)\widehat{\otimes }B(H)$ completely.