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

TL;DR

This paper investigates the embedding properties and ideal structure of the operator space projective tensor product, providing new insights into dual embeddings and the ideal lattice for certain operator algebras.

Contribution

It establishes a complete isometric embedding of the second dual tensor product into the dual of the tensor product and analyzes the ideal structure of the operator space projective tensor product.

Findings

Embedding of $V^{**}\otimes_h W^{**}$ into $(V\otimes_h W)^{**}$ is completely isometric.

Extension of bilinear forms from $V\times W$ to their second duals.

Complete description of the ideal structure of $A\hat{\otimes}B$ for $C^*$-algebras.

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.