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Let
be a right operator module over a
-algebra
,
a left
-operator module, and
an operator space.
A bilinear mapping
is called
balanced,
if the equation
obtains for all
.
The module Haagerup tensor product is defined to be the
operator space
(which is unique up to complete isometry) together with a
bilinear, completely contractive,
balanced mapping
such that the following holds true:
For each bilinear, completely bounded balanced map
there is a unique linear
completely bounded
map
satisfying
and
.
The module Haagerup tensor product can be realized in different ways:
- Let

.
The
quotient space
with its canonical matrix norms is an
operator space which satisfies the defintion
of
[BMP].
- Let us denote by
the algebraic module tensor product,
i.e. the quotient space
. For
and
, by
we define a semi-norm
on
.
We obtain
and the semi-norms
give an operator space norm on
[Rua89].
The completion of this space satisfies the definition of
[BMP].
Next: Examples
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Prof. Gerd Wittstock
2001-01-07