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

Abstract

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