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

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