Projective geometry in the Poincaré disk of a $C^*$-algebra

Abstract: 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.