TITLE: Is the space of *-homomorphisms between two $C^*$-algebras locally path connected
QUESTION [13 upvotes]: 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?

REPLY [4 votes]: If $A$ is semiprojective (the C*-analog of an absolute neighborhood retract, ANR), then your question also has a positive answer. The class of semiprojective C*-algebras includes also many non-commutative C*-algebras (e.g. all finite-dimensional algebras, Cuntz algebras).
More precisely, let $A$ be a semiprojective C*-algebra and let $B$ be any C*-algebra. Let $D=C([0,1],B)$ and consider the surjective homomorphism $\pi = ev_0\oplus ev_1\colon D\to B\oplus B$.
Given any homomorphism $f\colon A\to B$, consider the homomorphism $f\oplus f\colon a\mapsto (f(a),f(a)) \in B\oplus B$.
This has an obvious lift to a homomorphism $\alpha\colon A\to D$ such that $f\oplus f=\pi\circ\alpha$.
By Theorem 4.1 in Blackadar, "The homotopy lifting property for semiprojective C*-algebras", homomorphisms from semiprojective C*-algebras can be lifted if they are close enough to a liftable homomorphism. Let $g\colon A\to B$ be a homomorphism close to $f$. Then $f\oplus g$ is close to $f\oplus f$. Since the latter can be lifted to a homomorphism to $D$, so can $f\oplus g$. But this implies that $f$ is homotopic to $g$.