This is an announcement for the paper "Operator space projective tensor product: Embedding into second dual and ideal structure" by Ranjana Jain and Ajay Kumar.
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 $V^{**}\widehat{\otimes} W^{**}$ into $(V\widehat{\otimes} W)^{**}$ with a continuous inverse, where $\widehat{\otimes}$ is the operator space projective tensor product. Further, for $C^*$-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.
Archive classification: math.FA
Mathematics Subject Classification: 46L06, 46L07, 47L25
Remarks: 13 pages
Submitted from: rjain.math@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1106.2644
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