strictly positive elements under a nonzeo $*$ homomorphism

Question

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$B$?

Answer 1

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.