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

TL;DR

This paper explores the geometry of the Poincaré disk within a C*-algebra, introducing a cross ratio concept and linking it to the exponential map via geodesics, enriching the understanding of noncommutative geometric structures.

Contribution

It introduces the concept of cross ratio in the projective line of a C*-algebra and relates it to the exponential map and geodesics in the Poincaré disk setting.

Findings

Defined the cross ratio for four points in the projective line of a C*-algebra.

Established a relation between the exponential map and the cross ratio via geodesics.

Provided a geometric interpretation of the exponential map in the noncommutative setting.

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.