An operator space is called hilbertian, if is a Hilbert space . An operator space is called homogeneous, if each bounded operator is completely bounded and [Pis96].
Examples The minimal hilbertian operator space and the maximal hilbertian operator space , the column Hilbert space and the row Hilbert space are homogeneous hilbertian operator spaces on the Hilbert space .
Furthermore, for two Hilbert spaces and , we have completely isometric isomorphisms [ER91, Thm. 4.1] [Ble92b, Prop. 2.2]
These spaces satisfy the following dualities
[Ble92b, Prop. 2.2]
[Ble92a, Cor. 2.8]:
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For each Hilbert space there is a unique completely self dual homogeneous operator space, the operator Hilbert space [Pis96, §1]:
The intersection and the sum of two homogeneous hilbertian operator spaces are again homogeneous hilbertian operator spaces [Pis96].