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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]

$\displaystyle \mathit{CB}({\mathcal{C}}_{\H},{\mathcal{C}}_{\mathcal{K}}) \stackrel{\mathrm{cb}}{=}B(\H,\mathcal{K})$    and $\displaystyle \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]:

$\displaystyle {\mathcal{C}}_{\H}^*$ $\displaystyle \stackrel{\mathrm{cb}}{=}$ $\displaystyle {\mathcal{R}}_{\overline{\H}}$  
$\displaystyle {\mathcal{R}}_{\H}^*$ $\displaystyle \stackrel{\mathrm{cb}}{=}$ $\displaystyle {\mathcal{C}}_{\overline{\H}}$  
$\displaystyle \mathit{MIN}_{\H}^*$ $\displaystyle \stackrel{\mathrm{cb}}{=}$ $\displaystyle \mathit{MAX}_{\overline{\H}}$  
$\displaystyle \mathit{MAX}_{\H}^*$ $\displaystyle \stackrel{\mathrm{cb}}{=}$ $\displaystyle \mathit{MIN}_{\overline{\H}}$.  

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

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

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


next up previous contents index
Next: The morphisms Up: Hilbertian Operator Spaces Previous: Hilbertian Operator Spaces   Contents   Index
Prof. Gerd Wittstock 2001-01-07