Projective geometry in the Poincaré disk of a $C^{\ast }$-algebra

Abstract

We study the Poincaré disk $\mathcal{D}=\{a\in \mathcal{A}:\Vert a\Vert <1\}$ of a C${}^{\ast }$-algebra $\mathcal{A}$ from a projective point of view: $\mathcal{D}$ is regarded as an open subset of the projective line ${\mathbb{P}}_1\mathcal{A}$, the space of complemented rank one submodules of ${\mathcal{A}}^2$. We introduce the concept of cross ratio of four points in ${\mathbb{P}}_1\mathcal{A}$. Our main result establishes the relation between the exponential map $Ex{p}_{z_0}(z_1)$ 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.

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References in corpus (1)

• Poncaré half-space of a C*-algebra

Cited by in corpus (1)

• A non commutative Kähler structure on the Poincaré disk of a C*-algebra