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

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We prove that for operator spaces V and W, the operator space V**⊗ₕ W** can be completely isometrically embedded into (V⊗ₕ W)**, ⊗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× W can be extended uniquely to a separately w*-continuous jointly completely bounded bilinear form on V**× W**. This paves the way to obtain a canonical embedding of V**⊗ W** into (V⊗ W)** with a continuous inverse, where ⊗ is the operator space projective tensor product. Further, for C*-algebras A and B, we study the (closed) ideal structure of A⊗B, which, in particular, determines the lattice of closed ideals of B(H)⊗ B(H) completely.