Injective operator space tensor product

The representations of two operator spaces $X$ in $B(\H)$ and $Y$ in $B(\mathcal{K})$ yield a representation of the algebraic tensor product of $X$ and $Y$ in $B(\H\otimes_2 \mathcal{K})$. The operator space structure obtained in this way turns out to be independent of the representations chosen. It is called the injectiveoperator space tensor product of $X$ and $Y$ and is denoted by $X\stackrel{\scriptscriptstyle \vee}{\otimes}Y$ [BP91, p. 285]. Hence, in the case of $C^*$-algebras, the injective operator space tensor product and the minimal $C^*$-tensor product coincide.³³

By means of the duality of tensor products we obtain a formula [BP91, Thm. 5.1] for the injective operator space tensor norm of an element $u \in M_n(X\otimes Y)$ which is representation free:

$$\Vert u\Vert _\vee = \sup \Vert\langle u, \varphi \otimes \psi \rangle\Vert _{M_{nkl}},$$

where $k,l \in {\mathbb{N}}$, $\varphi \in \mathrm{Ball}(M_k(X^*))$ and $\psi \in \mathrm{Ball}(M_l(Y^*))$.

Interpreting, as is usual, the elements of the algebraic tensor product as finite rank operators we have the completely isometric embeddings [BP91, Cor. 5.2]

$$X\stackrel{\scriptscriptstyle \vee}{\otimes}Y \hookrightarrow \mathit{CB}(X^*,Y)$$   resp. $$\quad X \stackrel{\scriptscriptstyle \vee}{\otimes}Y \hookrightarrow \mathit{CB}(Y^*,X)$$.$$$$

The injective operator space tensor norm is the least cross norm whose dual norm again is a cross norm.

The injective operator space tensor product is symmetric , associative and injective . But it is not projective [BP91, Cor. 5.2].

The injective norm is the dual norm of the projective operator space tensor norm [BP91, Thm. 5.6]; but the projective operator space tensor norm is not in general the dual of the injective operator space tensor norm even if one of the two spaces involved is finite dimensional [ER90a, p. 168], [ER91, p. 264].

Footnotes

... coincide.³³

For this reason the injective operator space tensor product is also called spatial tensor product of operator spaces and denoted by $X \otimes_{\mathrm{min}} Y$ .