算子空间投射张量积：嵌入第二对偶与理想结构
Operator space projective tensor product: Embedding into second dual and ideal structure
摘要：
证明了对于算子空间$V$和$W$，算子空间$V^{**}\otimes_h W^{**}$可以完全等距嵌入$(V\otimes_h W)^{**}$，$\otimes_h$是Haagerup张量积.证明了对正合算子空间$V$和$W$，$V\times W$上的联合完全有界双线性型可以唯一地推广到$ V^{**}\times W^{**}$上的$w^*$ -连续的联合完全有界双线性型.这为获得具有连续逆的$V^{**}\hat{\otimes} W^{**}$到$(V\hat{\otimes} W)^{**}$的规范嵌入铺平了道路，其中$\hat{\otimes}$是算子空间投射张量积。进一步，对$C^*$ -代数$A$和$B$，我们研究了$A\hat{\otimes}B$的（闭）理想结构，特别是它完全决定了$B(H)\hat{\otimes} B(H)$的闭理想格.
英文摘要：
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.
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