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Let
,
be operator spaces. We say that a linear map
factors through a
column Hilbert space ,
if there is a Hilbert space
and completely bounded maps
,
with
.
We define

,
where the infimum runs over all possible factorizations.
If no such factorisation exists we say
.
is the Banach space of all linear
maps
with
[ER91, Chap. 5],[Ble92b, p. 83].
Let
and
be operator spaces
and
,
,
.
Then we have the
ideal property

.
We interpret a matrix
as a mapping
from
to
:
.
has a factorization in completely bounded mappings
Again, we define

,
where the infimum is taken over all factorizations.
So we get an operator space structure on
[ER91, Cor. 5.4].
Let
be operator spaces and
an
operator subspace of
.
Then the inclusion
is completely isometric [ER91, Prop. 5.2].
Let
,
be operator spaces. It is well known that
every linear map
defines a linear functional
via

.
This identification determines the complete isometry
[ER91, Thm. 5.3] [Ble92b, Thm. 2.11]

.
Let
,
,
be operator spaces.
We get a complete isometry
via the mapping
[ER91, Cor. 5.5].
Next: Multiplicative Structures
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Prof. Gerd Wittstock
2001-01-07