Given the set of *-homomorphisms between two $C^*$-algebras $A$ and $B$, we may define a metric on it by setting $d(f,g):= \sup_{0<\|a\|\le 1}\|f(a)-g(a)\|$. Could it be true that if $d(f,g)<\epsilon$ for some $\epsilon$ small enough, then there exists a homotopy between $f$ and $g$, i.e. a *-homomorphism $H\colon A\to C([0;1];B)$ such that $ev_0H=f$ and $ev_1H=g$? If not, could it be true when $B$ is stable?

Given the set of *-homomorphisms between two $C^*$-algebras $A$ and $B$, we may define a metric on it by setting $d(f,g):= \sup_{0<\|a\|\le 1}\|f(a)-g(a)\|$. Could it be true that, for each *-homomorphism $f\colon A \to B$ there exists such $\epsilon >0$ that if $d(f,g)<\epsilon$, then there exists a homotopy between $f$ and $g$, i.e. a *-homomorphism $H\colon A\to C([0;1];B)$ such that $ev_0H=f$ and $ev_1H=g$? If not, could it be true when $B$ is stable?