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| COMPLETELY POSITIVE MAPS AND$*$-OMORPHISM OF $\[{C^*}\]$-ALGEBRAS |
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Citation： |
Wu Liangsen.COMPLETELY POSITIVE MAPS AND$*$-OMORPHISM OF $\[{C^*}\]$-ALGEBRAS[J].Chinese Annals of Mathematics B,1988,9(1):27~31 |
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Net amount： 748 |
Authors： |
Wu Liangsen; |
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| Abstract： |
Let $A$, $B$ be unital $\[{C^*}\]$-algebras.
$\[{\chi _A} = \{ \varphi |\varphi \]$ are all completely postive linear maps from $\[{M_n}(C)\]$ to $A$ with $\[\left\| {a(\varphi )} \right\| \le 1\]$ $}$.
$\[(a(\varphi ) = \left( {\begin{array}{*{20}{c}}
{\varphi ({e_{11}})}& \cdots &{\varphi ({e_{1n}})}\{}& \cdots &{}\{\varphi ({e_{n1}})}& \cdots &{\varphi ({e_{nn}})}
\end{array}} \right),\]$ where $\[\{ {e_{ij}}\} \]$ is the matrix unit of $\[{M_n}(C)\]$.
Let $\[\alpha \]$ be the natural action of $\[SU(n)\]$ on $\[{M_n}(C)\]$
For $\[n \ge 3\]$, if $\[\Phi \]$ is an $\[\alpha \]$-invariant affine isomorphism between $\[{\chi _A}\]$ and $\[{\chi _B}\]$, $\[\Phi (0) = 0\]$, then $A$ and $B$ are $\[^*\]$-isomorphic
In this paper a counter example is given for the case $\[n = 2\]$. |
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