Projective geometry in the Poincar\'e disk of a $C^{\ast }$-algebra

We study the Poincar'e disk $\mathcal{D}=\{a\in \mathcal{A}:\mathrm{\parallel }a\mathrm{\parallel }<1\}$ of a C${}^{\ast }$-algebra $\mathcal{A}$ from a projective point of view: $\mathcal{D}$ is regarded as an open subset of the projective line ${\mathbb{P}}_1\mathcal{A}$, the space of complemented rank one submodules of ${\mathcal{A}}^2$. We introduce the concept of cross ratio of four points in ${\mathbb{P}}_1\mathcal{A}$. Our main result establishes the relation between the exponential map $Ex{p}_{z_0}(z_1)$ of $\mathcal{D}$ ($z_0,z_1\in \mathcal{D}$) and the cross ratio of the four-tuple $$\delta (-\mathrm{\infty }),\delta (0)=z_0,\delta (1)=z_1,\delta (+\mathrm{\infty }),$$ where $\delta$ is the unique geodesic of $\mathcal{D}$ joining $z_0$ and $z_1$ at times $t=0$ and $t=1$, respectively.