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The space
of completely bounded mappings between
two homogeneous hilbertian operator spaces enjoys
the following properties
(cf. [MP95, Prop. 1.2]):

is a Banach space.

for all ,
,
.

for all with
.
Consequently,
is a
symmetrically normed ideal (s.n. ideal) in the sense of
Calkin, Schatten [Sch70] and Gohberg [GK69].
The classical examples for s.n. ideals are the famous
Schatten ideals:
Many, but not all s.n. ideals can be represented as spaces of completely bounded maps
between suitable homogeneous hilbertian operator spaces.
The first result in this direction
was
isometrically [ER91, Cor. 4.5].
We have the following characterizations
isometrically or only isomorphically ()
[Mat94], [MP95], [Lam97]:
As a unique completely isometric isomorphism, we get
(cf. [Ble95, Thm. 3.4]).
The result
is of special interest. Here, we have a new quite natural norm on the
nuclear operators,
which is not equal to the canonical one.
Even in the finite dimensional case, we only know
[Pau92, Thm. 2.16].
To compute the exact constant is still an open problem.
Paulsen conjectured that the upper bound is sharp [Pau92, p. 121].
Let be a Banach space and a Hilbert space.
An operator
is completely bounded from
to
,
if and only if is 2summing
[Pie67] from to
(cf. [ER91, Thm. 5.7]):
with
.
Next: The column Hilbert space
Up: Hilbertian Operator Spaces
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Prof. Gerd Wittstock
20010107