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For matricially normed spaces
and
,
,
and
we have
and

.
A matrix
defines an operator
Thus we have an algebraic identification of
and
and
further of
and
. The latter even is a complete isometry
([Ble92b, Cor. 2.14]):

.
The isometry on the first matrix level is shown in [Ble92a, Thm. 2.5]. This
already implies11the complete isometry.
More generally we have12

.
is called reflexive,
if
. An
operator space
is reflexive if and only if its first matrix level
is a reflexive Banach space.
Footnotes
- ... implies11
-

.
- ... have12
-
This follows from
and the above mentioned formula

.
Next: The adjoint operator
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Prof. Gerd Wittstock
2001-01-07