The spaces

An operator space $X$ is called hilbertian, if $M_1(X)$ is a Hilbert space $\H$. An operator space $X$ is called homogeneous, if each bounded operator $T:M_1(X) \rightarrow M_1(X)$ is completely bounded and $\Vert T\Vert _{\mathrm{cb}}=\Vert T\Vert$ [Pis96].

Examples The minimal hilbertian operator space $\mathit{MIN}_{\H}:=\mathit{MIN}(\H)$ and the maximal hilbertian operator space $\mathit{MAX}_{\H}:=\mathit{MAX}(\H)$, the column Hilbert space ${\mathcal{C}}_{\H} := B({\mathbb{C}},\H)$ and the row Hilbert space ${\mathcal{R}}_{\H} :=B(\overline{\H},{\mathbb{C}})$ are homogeneous hilbertian operator spaces on the Hilbert space $\H$.

Furthermore, for two Hilbert spaces $\H$ and $\mathcal{K}$, we have completely isometric isomorphisms [ER91, Thm. 4.1] [Ble92b, Prop. 2.2]

$$\mathit{CB}({\mathcal{C}}_{\H},{\mathcal{C}}_{\mathcal{K}}) \stackrel{\mathrm{cb}}{=}B(\H,\mathcal{K})$$    and $$\mathit{CB}({\mathcal{R}}_{\H},{\mathcal{R}}_{\mathcal{K}}) \stackrel{\mathrm{cb}}{=}B(\overline{\mathcal{K}},\overline{\H}).$$

These spaces satisfy the following dualities [Ble92b, Prop. 2.2] [Ble92a, Cor. 2.8]:

$${\mathcal{C}}_{\H}^* \stackrel{\mathrm{cb}}{=} {\mathcal{R}}_{\overline{\H}}$$

$${\mathcal{R}}_{\H}^* \stackrel{\mathrm{cb}}{=} {\mathcal{C}}_{\overline{\H}}$$

$$\mathit{MIN}_{\H}^* \stackrel{\mathrm{cb}}{=} \mathit{MAX}_{\overline{\H}}$$

$$\mathit{MAX}_{\H}^* \stackrel{\mathrm{cb}}{=} \mathit{MIN}_{\overline{\H}}$$

For each Hilbert space $\H$ there is a unique completely self dual homogeneous operator space, the operator Hilbert space $\mathit{OH}_{\H}$ [Pis96, §1]:

$$\mathit{OH}_{\H}^* \stackrel{\mathrm{cb}}{=}\mathit{OH}_{\overline{\H}}$$.

The intersection and the sum of two homogeneous hilbertian operator spaces are again homogeneous hilbertian operator spaces [Pis96].