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Let
be a normed space. Among all operator space norms on
which coincide on the first matrix level with the given
norm, there is a greatest and a smallest. The matricially normed spaces given by these
are called
and
. They are characterized by the following
universal mapping property:14
For a matricially normed space
,
and
holds isometrically.
We have [Ble92a]
For
,
is not completely bounded[Pau92, Cor. 2.13].15
Subspaces of
-spaces are
-spaces: For each isometric mapping
, the mapping
is completely isometric.
Quotients of
-spaces are
-spaces: For each quotient mapping
, the mapping
is a complete quotient mapping.
Footnotes
- ... property:14
-
is the left adjoint and
the right adjoint of the forgetfull functor which
maps an operator space
to the Banach space
.
- ...Paulsen92.15
-
Paulsen uses in his proof a false estimation for the projection constant of the finite
dimensional Hilbert spaces; the converse estimation is correct
[Woj91, p. 120], but here useless.
The gap can be filled [Lam97, Thm. 2.2.15]
using the famous theorem of Kadets-Snobar:
The projection constant of an
-dimensional Banach space is less or equal
than
[KS71].
Subsections
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Prof. Gerd Wittstock
2001-01-07