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?
Is the space of *-homomorphisms between 2 $C^*$-algebras locally path connected
Kolya Ivankov
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