functional calculus under the $*$ homomorphism

Question

If $A,B$ are two $C^*$ algebras,$\psi:A \rightarrow B$ is a non $*$ homomorphism.Suppose $b$ is a nonzero normal element in $B$,we have a $*$ isometric isomorphism $\phi:C(\sigma_B(b))\to C^*(b,b^*),\; f\mapsto f(b)$

My question is:does there exist a $f\in C(\sigma_B(b)),a\in A$ such that $\psi(a) =f(b)$ is nonzero?

Answer 1

This can fail even if $\psi$ is a $*$-homomorphism. Take $A=B=\mathbb C\oplus\mathbb C$, and $$ \psi(x,y)=(x,0),\ \ \ b=(0,1). $$ Then $\psi(A)=\mathbb C\oplus 0$, while $C^*(b)=0\oplus\mathbb C$; so your equality can only occur when $\psi(a)=0=f(b)$.