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Let , be Banach spaces.
We consider a linear subspace of the space
of the continuous operators between and
which contains all finite rank maps
and is a Banach space with respect to a given norm.
Furthermore, it is usually required that be defined for all pairs
of Banach spaces and .
Such a space is called - according to A. Grothendieck - a
mapping space
.
Analogously, we call an operator space
which is a linear subspace of
a
-mapping space.
Note that generally the algebraic identification of
with
fails to be isometric and that
the norms on
do not generate an
operator space structure for .
There is a close relationship between mapping spaces and
tensor products:
The space of all finite rank maps
between and
and the algebraic tensor product of with are isomorphic:
This identification enables us to transfer norms from one space
to the other one.
To this end, we consider the extension of the mapping
to the completion with respect to an operator space tensor norm
:
.
is in general neither injective nor surjective.
As a CB-mapping space one obtains
with the operator space
norm of
.
We consider now assignments that assign a
mapping space
with operator space norm
to every pair of
operator spaces. In the Banach space theory
A. Pietsch intensified the notion of mapping spaces to that of
the
operator ideals [Pie78].
Analogously, we consider operator ideals which are
mapping spaces with the
-ideal property [ER94],
i.e , the composition
is for all operator spaces , , , defined and
jointly completely contractive .
A
-ideal is called local [EJR98],
if its norm satisfies:
.
Subsections
Next: Completely nuclear mappings
Up: What are operator spaces?
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Prof. Gerd Wittstock
2001-01-07