# strictly positive elements under a nonzeo $*$ homomorphism

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$$B$$?

• 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. – user42761 Dec 5 '18 at 11:32

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.