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Construction of $ \mathit{MIN}$:

For a commutative $ C^*$-algebra $ A=C(K)$, each bounded linear mapping $ \Phi:M_1(X)\rightarrow A$ is automatically completely bounded with $ \Vert\Phi\Vert _{\mathrm{cb}}=\Vert\Phi\Vert$ [Loe75].16

Each normed space $ E$ is isometric to a subspace of the commutative $ C^*$-algebra $ l_\infty(\mathrm{Ball}(E^*))$. Thus the operator space $ \mathit{MIN}(E)$ is given as a subspace of $ l_\infty(\mathrm{Ball}(E^*))$.

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

$\displaystyle \Vert x\Vert=\sup\left\{\left. \Vert f^{(n)}(x)\Vert \right\vert f\in\mathrm{Ball}(E^*)\right\}$   .

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



Footnotes

...Loebl75.16
I. e. $ A$ is a $ \mathit{MIN}$-space.


Prof. Gerd Wittstock 2001-01-07