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Tensor products of operator matrices
As usual we define the algebraic tensor product of operator matrices
,
by setting
Here we have used the definition
and the associative law
 |
(5) |
In view of the next identification one should note that the
shuffle-map is an algebraic isomorphism:
 |
(6) |
for
,
,
.
The shuffle-isomorphism at hand we obtain the identification:
Finally we use the usual72
identification
to obtain
 |
(7) |
We call this algebraic isomorphism the
shuffle-isomorphism.
One should note that for operator space tensor products
the algebraic identifications
(
) and (
)
are only complete contractions:
In general these are not isometries even for
resp.
.
For an operator space tensor product the
shuffle-map
 |
(10) |
in general is
only
completely contractive.
In the case of the
injective operator space tensor product
this is of course a complete isometry.
More generally, one considers the shuffle-map for
rectangular matrices73:
Another example is provided by the Blecher-Paulsen
equation .
Footnotes
- ... usual72
- In the matrix so obtained
are the row indices and
are the
column indices, where
and
.
The indices
resp.
are ordered lexicographically.
- ... matrices73
- The shuffle-map
,
,
,
operator spaces, has been studied for various combinations of
operator space tensor products
[EKR93, Chap. 4].
Next: Joint amplification of a
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Prof. Gerd Wittstock
2001-01-07