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Matrices over an operator space
The vector space
of
matrices over a matricially normed space
itself is matricially
normed in a natural manner: The norm on the
th level
is given by the identification
[BP91, p. 265].
We8write
for
with this operator space structure.
In particular,
holds isometrically.
Analogously
becomes a matricially normed space
by the identification

.
By adding zeros it is a subspace of
for
,
.
Examples:
For a
-algebra
,
is the
-Algebra of
-matrices over
with its
natural operator space structure.
The Banach space
is the first matrix level of the
operator space
.
The complex numbers have a unique operator space structure which on the first
matrix level is isometric to
, and for this
holds isometrically.
We write

.
Then
always stands for the
-algebra of
-matrices with its
operator space structure. The Banach space
is the first matrix level
of the operator space
.
Footnotes
- ...
We8
-
In the literature, the symbol
stands for both the operator space with first
matrix level
and for the
th level of the operator space
.
We found that the distinction between
and
clarifies for instance the definition of the operator space
structure of
.
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Prof. Gerd Wittstock
2001-01-07