The column Hilbert space ${\mathcal{C}}_\H$

For the Hilbert space $\H=\ell_2$, we can realize the column Hilbert space ${\mathcal{C}}_\H$ as a column, by the embedding

$$\begin{array}{ccc} \H& \hookrightarrow & B(\H), \\ \left... ...s & \vdots & & & \ddots \end{array} \right]\mbox{.} \end{array}$$

Via this identification, ${\mathcal{C}}_{\ell_2^n}$ is the $n$-dimensional column space ${\mathcal{C}}_n$.

${\mathcal{C}}_\H$ is a homogeneous hilbertian operator space : All bounded maps on $\H$ are completely bounded with the same norm on ${\mathcal{C}}_\H$. Actually we have $\mathit{CB}({\mathcal{C}}_\H) \stackrel{\mathrm{cb}}{=}B(\H)$ completely isometrically [ER91, Thm. 4.1].

${\mathcal{C}}_\H$ is a injective operator space (cf. [Rob91]).