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Let $A, B$ be the C*-algebra. Assume $A$ is nonunital, $B$ is unital and $\phi: A \rightarrow B$ is a contractive completely positive map.

Then we consider the double adjoint map $\phi^{**}: A^{**}\rightarrow B^{**}$. Identifying double duals with enveloping von Neumann algebras, can we checks that $\phi^{**}$ maps positive operators to positive operators?

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The key observation is that the identification between $A^{**}$ and the enveloping von Neumann algebra preserves positivity. So, if $\alpha\in A^{**}_+$, we can find a net $\{a_n\}\in A_+$ with $a_n\to\alpha$ in the $w^*$-topology (every $a\in A''_+$ is a weak-limit of elements in $A_+$).

Now let $f\in B^*_+$. Then $$ (\phi^{**}\alpha)f=\alpha(\phi^*f)=\lim_na_n(\phi^*f)=\lim_n f(\phi(a_n))\geq0, $$ using that $\phi$, $f$ are positive.

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  • $\begingroup$ I still have a simple question why does $(\phi^{**}\alpha)f=\alpha(\phi^*f)$ hold? $\endgroup$ – Yan kai Mar 25 '14 at 4:32
  • $\begingroup$ By definition of $\phi^{**}$. More precisely, by definition of the adjoint map. $\endgroup$ – Martin Argerami Mar 25 '14 at 4:34

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