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Let
and
be operator spaces such that
and
are embedded in
a Hausdorff topological vector space.
is given a norm via
.
So we have:
![$\displaystyle { \left\Norm [x_{ij}] \right\Norm }_{M_n(X \cap Y)}
= \max \left...
...\right\Norm }_{M_n(X)} , { \left\Norm [x_{ij}] \right\Norm }_{M_n(Y)}
\right\}$](img1365.png)
.
The operator space
is called the
intersection of
and
.
For operator spaces75
and
, by embedding
in
we obtain an operator space structure
.
We write
.
The quotient operator space
is called the
sum of
and
and is denoted by
.
We have
![$\displaystyle { \left\Norm [x_{ij}] \right\Norm }_{M_n(X+Y)}
= \inf_{[x_{ij}] ...
...{ \left\Norm [\left(x_{ij_X},x_{ij_Y}\right)] \right\Norm }_{M_n(X \oplus_1 Y)}$](img1377.png)
.
Footnotes
- ... spaces75
-
Let
,
be Banach spaces. Then we have their 1-direct sum
with the norm
and their sum
with the quotient norm
Prof. Gerd Wittstock
2001-01-07