# Journal of Operator Theory

Volume 89, Issue 1, Winter 2023  pp. 155-182.

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

Authors:  E. Andruchow (1), G. Corach (2), L. Recht (3)
Author institution: (1) Instituto de Ciencias, Universidad Nacional de General Sarmiento, Los Polvorines, 1613, Argentina and Instituto Argentino de Matem\'atica "Alberto P. Calderon", CONICET, Buenos Aires, 1083, Argentina
(2) Instituto Argentino de Matematica "Alberto P. Calderon", CONICET, Buenos Aires, 1083, Argentina
(3) Instituto Argentino de Matematica "Alberto P. Calderon", CONICET, Buenos Aires, 1083, Argentina

Summary: We study the Poincar\'e disk ${\mathcal D} =\{a\in\mathcal A : \|a\|<1\}$ of a $C^*$-algebra $\mathcal A$ from a projective point of view: ${\mathcal D}$ is regarded as an open subset of the projective line $\mathcal A\mathbb{P}$, the space of complemented rank one submodules of $\mathcal A ^2$. We introduce the concept of cross ratio of four points in $\mathcal A\mathbb{P}$. Our main result establishes the relation between the exponential map $\mathrm{Exp}_{z_0}(z_1)$ of ${\mathcal D}$ ($z_0,z_1\in{\mathcal D}$) 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 ${\mathcal D}$ joining $z_0$ and $z_1$ at times $t=0$ and $t=1$, respectively. Here $\delta(-\infty)=\lim\limits_{t\to-\infty}\delta(t)$ and $\delta(+\infty)=\lim\limits_{t\to+\infty}\delta(t)$, the limits are considered in the strong operator topology, and may take values in the universal algebra $\mathcal A ^{**}$.

DOI: http://dx.doi.org/10.7900/jot.2021dec27.2356
Keywords: projective line, Poincar\'e disk, $C^*$-algebra