Let $A$ and $B$ be unital $C^*$-algebras and suppose that there is a surjective *-homomorphism $f:A\rightarrow B$. Then any invertible element in $B$ that is connected to $1_B$ can be lifted to an invertible element in $A$ that is connected to $1_A$ since such elements are products of exponentials.
I am considering the following situation where some restrictions are imposed. Assume that $A=\overline{\bigcup_{r\geq 0}A_r}$ where
- each $A_r$ is a linear subspace of $A$ containing $1_A$;
- each $A_r$ is closed under involution;
- $A_r\subset A_{r'}$ if $r\leq r'$;
- $A_r A_{r'}\subset A_{r+r'}$ for all $r,r'$.
Then $(f(A_r))_{r\geq 0}$ forms a collection of subspaces with the same properties with respect to $B$. Suppose that $u\in f(A_r)$ is such that $||uu^*-1||<\varepsilon$, $||u^*u-1||<\varepsilon$, and $u$ is connected to $1_B$ by a path of elements with the same properties. Does there exist some constants $\alpha,\beta$, some integer $j$, and some $v\in M_{j+1}(A_{\beta r})$ such that $||vv^*-1||<\alpha\varepsilon$, $||v^*v-1||<\alpha\varepsilon$, $v$ is connected to $I$, and $||f(v)-\mathrm{diag}(u,I_j)||<\alpha\varepsilon$? (Here I am using $f$ to denote the induced homomorphism between the matrix algebras.)
Some other assumption may be needed, for example having a linear map $s:B\rightarrow A$ with $s(1)=1$ and $f\circ s=id$.