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Notations

Using the disjoint union

$\displaystyle M(X):=\dot{\bigcup_{n\in{\mathbb{N}}}}M_n(X)$   ,

the notation becomes simpler.7



Footnotes

... simpler.7
The norms on the matrix levels $ M_n(X)$ are then one mapping $ M(X)\to{\mathbb{R}}$. The amplifications of $ \Phi:X\to Y$ can be described as one mapping $ \Phi:M(X)\to M(Y)$. We have

$\displaystyle \Vert\Phi\Vert _{\mathrm{cb}}=\sup\{\Vert\Phi(x)\Vert\;\vert\;x\in M(X),\;\Vert x\Vert\leqslant 1\}$.

$ \Phi$ is completely isometric if $ \Vert x\Vert=\Vert\Phi(x)\Vert$ for all $ x\in M(X)$, and $ \Phi$ is a complete quotient mapping if $ \Vert y\Vert=\inf\{\Vert x\Vert\;\vert\;x\in\Phi^{-1}(y)\}$ for all $ y\in M(Y)$ or $ \Phi(\mathrm{Ball}^\circ X)=\mathrm{Ball}^\circ Y$, where $ \mathrm{Ball}^\circ X = \{x\in M(X)\;\vert\;\Vert x\Vert< 1\}$.


Prof. Gerd Wittstock 2001-01-07