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In the literature, there are two different notions of an
amplification
of a bilinear mapping.
We shall call the first kind
of amplification the
joint
amplification.
This joint amplification is needed to obtain a matrix duality
- which is
fundamental in the duality theory of operator spaces -,
starting from an ordinary duality
.
The notion of joint amplification leads to the
jointly completely bounded
bilinear maps as well as the
projective operator space tensor product.
We will speak of the second kind
of an amplification as the
amplification
of a bilinear mapping.
This notion leads to the
completely bounded
bilinear maps and the
Haagerup tensor product.
In the sequel, we will use the notation
for a bilinear mapping and
for its linearization.
Both notions of an amplification of a bilinear map
are formulated in terms of the
amplification of its linearization:
-
The
joint amplification of
produces
the bilinear mapping
of the operator matrices
and
.
Here,
the tensor product of operator matrices is
defined
via
-
In
the case of
completely bounded
bilinear maps one deals with the
tensor
matrix multiplication
[Eff87]
of operator matrices
and
.
For more formulae, see:
tensor matrix multiplication .
The (n,l)-th
amplification
of a bilinear map
is defined by
for
,
,
.
In case
, we shortly write48
Footnotes
- ... write48
- In the literature, the amplification of a bilinear map often is defined only for
quadratic operator matrices and is called
the amplification
.
Nevertheless,
we will also be dealing with the amplification for rectangular matrices
since this permits
the formulation of
statements
where for fixed
all amplifications
,
, are considered.
Next: Jointly complete boundedness
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Prof. Gerd Wittstock
2001-01-07