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Suppose $A$ is a separable $C^*$ algebra,x is a strictly positive element in $A$,$\phi:A\rightarrow B$ is a nonzero $*$ homomorphism,is $\phi(x)$ also strictly positive in$B$?

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  • $\begingroup$ In general, if $A \subseteq B$ and $a \in A$ is strictly positive, then $a$ might not be strictly postive in $B$, e.g. non-unital inclusions. $\endgroup$ – user42761 Dec 5 '18 at 11:32
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No. Take $\phi:C([0,1])\to C([0,1]\cup\{2\})$ to be the map such that $(\phi f)(t)=f(t)$ whenever $t\in[0,1]$ and $(\phi f)(2)=0$ for all $f\in C([0,1])$. Then any element in the image of $\phi$ is non-invertible, hence not strictly positive.

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