Projective geometry of the Poincaré disk of a C*-algebra

Abstract

We study the Poincar´e disk $d={aina: |a|<1}$ of a C$^*$-algebra $a$ from a projective point of view: $d$ is regarded as an open subset of the projective line $pa$, the space of complemented rank one submodules of $a^2$. We introduce the concept of cross ratio of four points in $pa$. Our main result establishes the relation between the exponential map $Exp_{z_0}(z_1)$ of $d$ ($z_0,z_1ind$) 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 $d$ joining $z_0$ and $z_1$ at times $t=0$ and $t=1$, respectively. Here $delta(-infty)=lim_{to-infty}delta(t)$ and $delta(+infty)=lim_{to+infty}delta(t)$, the limits are considered in the strong operator topology, and may take values in the universal algebra $a^{**}$.