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Construction of $ \mathit{MAX}$:

For a index set $ I$, $ l_1(I) = c_0(I)^*$. $ l_1(I)$ is an operator space as dual of the commutative $ C^*$-algebra $ c_0(I)$, and each bounded linear mapping $ \Phi:l_1(I)\rightarrow M_1(X)$ is automatically completely bounded with $ \Vert\Phi\Vert _{\mathrm{cb}}=\Vert\Phi\Vert$.17

Each Banach space18$ E$ is isometric to a quotient of $ l_1(\mathrm{Ball}(E))$. Thus the operator space $ \mathit{MAX}(E)$ is given as a quotient of $ l_1(\mathrm{Ball}(E))$.

For $ x\in M_n(\mathit{MAX}(E))$ we have

$\displaystyle \Vert x\Vert=\sup\{\Vert\varphi^{(n)}(x)\Vert\;\vert\;n\in {\mathbb{N}},\;
\varphi:E\to M_n,\;\Vert\varphi\Vert\leqslant 1\}$.

The unit ball of $ \mathit{MAX}(E)$ is given as the absolute matrix bipolar of $ \mathrm{Ball}(E^)$.

Footnotes

....17
I. e. $ l_1(I)$ is a $ \mathit{MAX}$-space.
... space18
A similar construction is possible for normed spaces.


Prof. Gerd Wittstock 2001-01-07