Let me thoroughly answer this question in the commutative case. Hopefully this is helpful for the case when the $C^{*}$-algebra is not necessarily commutative.

If $B$ is a commutative $C^{*}$-algebra, then $B$ is isomorphic to $C(X)$ for some compact Hausdorff space $X$. Here, $X$ is simply the collection of all maximal $^{*}$-ideals of the $C^{*}$-algebra $B$.

Furthermore, if $B$ is generated by an element $a$, then $B$ is isomorphic to $C(\sigma(a))$ by an isomorphism that maps $a$ to the inclusion mapping $\iota:\sigma(a)\rightarrow\mathbb{C}$.

Let $X$ be a compact Hausdorff space. Suppose therefore that $g,h\in C(X)$ are invertible elements. Then there is a path from $g$ to $h$ in $C(X)^{\times}$ if and only if $g,h:X\rightarrow\mathbb{C}\setminus\{0\}$ are homotopic. Let $\pi:\mathbb{C}\setminus\{0\}\rightarrow S^{1}$ be the function defined by $\pi(z)=\frac{z}{|z|}$. Then $\pi$ is a homotopy equivalence, so
there is a path from $g$ to $h$ in $C(X)^{\times}$ if and only if
$\pi\circ g$ and $\pi\circ h$ are homotopic to each other.

If $X$ is a topological spaces, then observe that the collection of mappings $f:X\rightarrow S^{1}$ is an abelian group where
$(f\cdot g)(x)=f(x)\cdot g(x)$. Homotopy equivalence is a congruence on this abelian group, so the collection of homotopy classes $[X,S^{1}]$ of mappings from $X$ to $S^{1}$ is also an abelian group. One might suspect that there is a relationship between $[X,S^{1}]$ and some sort of homology, cohomology, or homotopy groups, and there is.

It turns out that if $X$ is homotopy equivalent to a CW complex, then the group $[X,S^{1}]$ is isomorphic to the singular cohomology group
$H^{1}(X,\mathbb{Z})$. If $X$ is not homotopy equivalent to a CW complex, then we will have to use a different notion of cohomology. Fortunately, it has been shown by Peter Huber [1] that for paracompact Hausdorff spaces $X$, the group $[X,S^{1}]$ is isomorphic to the Cech-cohomology group
$\check{H}^{1}(X,\mathbb{Z})$. These are actually special cases of more general isomorphisms $H^{n}(X,G)\simeq[X,K(G,n)]$ where $K(G,n)$ is the Eilenberg-MacLane spaces where $X$ is homotopy equivalent to a CW-complex or the isomorphism $\check{H}^{n}(X,G)\simeq[X,K(G,n)]$ where $X$ is paracompact and Hausdorff and $G$ is countable. The special case follows since $S^{1}=K(\mathbb{Z},1)$.

- Huber, P.J. Homotopical Cohomology and čech Cohomology. Math. Ann. 144, 73–76 (1961). https://doi.org/10.1007/BF01396544