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cross norms

Sometimes one considers an operator space norm on the algebraic tensor product of two fixed operator spaces. Then one usually demands that this norm and its dual norm are at least cross norms. Operator space tensor norms always have these properties.

An operator space norm $ \Vert\cdot\Vert _\alpha $ on the algebraic tensor product $ X \otimes Y $ of two operator spaces $ X$ and $ Y$ is said to be a cross norm, if

$\displaystyle \Vert x \otimes y\Vert _{\alpha,pq}=\Vert x\Vert _p\, \Vert y\Vert _q
$

for all $ p,q\in{\mathbb{N}}$, $ x \in M_p(X) $, $ y\in M_q(Y) $ holds.

For cross norms $ {\mathbb{C}}\otimes_\alpha X \stackrel{\mathrm{cb}}{=}X $ is completely isometric.

For an operator space norm $ \Vert\cdot\Vert _\alpha $ on the algebraic tensor product of two fixed operator spaces $ X$ and $ Y$ one usually asks for the following three properties (i)-(iii).32

(i)
$ \Vert\cdot\Vert _\alpha $ is a cross norm.

(ii)
Let $ \varphi \in X^* $, $ \psi \in Y^* $ be linear functionals and
$\displaystyle \varphi \otimes \psi : X \otimes Y$ $\displaystyle \rightarrow$ $\displaystyle {\mathbb{C}}$  
$\displaystyle \langle x \otimes y, \varphi \otimes \psi \rangle$ $\displaystyle :=$ $\displaystyle \langle x,\varphi \rangle \langle y,\psi \rangle$  

where $ x \in X$, $ y \in Y $. their tensorproduct. The tensor product $ \varphi \otimes \psi $ has a continuous linear extension to $ X \otimes_\alpha Y $.

Then the dual operator space norm $ \Vert\cdot\Vert _{\alpha^*} $ is defined on the algebraic tensor product $ X^ * \otimes Y^* $ by the algebraic embedding

$\displaystyle X \otimes Y \subset (X^* \otimes_\alpha Y^*)^*.
$

(iii)
The dual operator space norm $ \Vert\cdot\Vert _{\alpha^*} $ is a cross norm.

There is a smallest operator space norms among the operator space norms on $ X \otimes Y $, for which $ \Vert\cdot\Vert _\alpha $ and the dual norm $ \Vert\cdot\Vert _{\alpha^*} $ are cross norms. This is the injective operator space tensor norm $ \Vert\cdot\Vert _\vee $. [BP91, Prop. 5.10].

There is a greatest operator space norm among the operator space norms on $ X \otimes Y $, for which $ \Vert\cdot\Vert _\alpha $ and the dual norm $ \Vert\cdot\Vert _{\alpha^*} $ are cross norms. This is the projective operator space tensor norm $ \Vert\cdot\Vert _\wedge $ [BP91, Prop. 5.10].

On the algebraic tensor product $ X \otimes Y $ one can compare the operator space norms $ \Vert\cdot\Vert _\alpha $ for which $ \Vert\cdot\Vert _\alpha $ and the dual norm $ \Vert\cdot\Vert _{\alpha^*} $ are cross norms with the injective tensor norm $ \Vert\cdot\Vert _\lambda $ and the projective tensor norm $ \Vert\cdot\Vert _\gamma $ of normed spaces:

$\displaystyle \Vert\cdot\Vert _\lambda \leq
\Vert\cdot\Vert _{\vee,1} \leq
\Ve...
...rt _{\alpha,1} \leq
\Vert\cdot\Vert _{\wedge,1} \leq
\Vert\cdot\Vert _\gamma.
$



Footnotes

...32
The conditions (i)-(iii) are equivalent to the following: The bilinear maps
$\displaystyle X \times Y \rightarrow X \otimes_\alpha Y,$   $\displaystyle (x,y) \mapsto x \otimes y$  
$\displaystyle X^* \times Y^* \rightarrow (X \otimes_\alpha Y)^*,$   $\displaystyle (\varphi,\psi) \mapsto \varphi \otimes \psi$  

are jointly completely bounded .

next up previous contents index
Next: Injective operator space tensor Up: Operator space tensor products Previous: Operator space tensor products   Contents   Index
Prof. Gerd Wittstock 2001-01-07