Is $h^{-1}(C)$ a C*-algebra if $h\colon A\to B$ is a $\ast$-homomorphism between C*-algebras and $C\subset B$ is a C*-subalgebra?

Question

Is $h^{-1}(C)$ a C*-algebra if $h\colon A\to B$ is a $\ast$-homomorphism between C*-algebras and $C\subset B$ is a C*-subalgebra?

I know the image of a C*-subalgebra under a $\ast$-homomorhism is again a C*-algebra, but is it also true for preimages?

Answer 1

Yes. Because $h$ is continuous, the preimage of a closed set is closed. So $h^{-1}(C)$ is closed. That it is a $*$-algebra follows easily from the fact that $h$ is s $*$-homomorphism.