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 $V^{\ast \ast }{\otimes }_h{W}^{\ast \ast }$ 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 $V^{\ast \ast }\times W^{\ast \ast }$. This paves the way to obtain a canonical embedding of $V^{\ast \ast }\hat{\otimes }{W}^{\ast \ast }$ into $(V\hat{\otimes }W{)}^{\ast \ast }$ with a continuous inverse, where $\hat{\otimes }$ is the operator space projective tensor product. Further, for $C^{\ast }$-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.

13 pages