$C^*$ -代数的Poincaré圆盘中的Projective geometry
Projective geometry in the Poincaré disk of a $C^*$-algebra
摘要：
从射影的角度研究了C-$^*$ -代数${\cal A}$的Poincaré圆盘${\cal D}=\{a\in {\cal A}: \|a\|<1\}$：${\cal D}$被看作是${\cal A}^2$的补秩1子模空间--射影线$\mathbb{P}_1{\cal A}$的开子集.在$\mathbb{P}_1{\cal A}$中引入了四点交比的概念。我们的主要结果建立了${\cal D}$的指数映射$Exp_{z_0}(z_1)$（$z_0,z_1\in {\cal D}$）和四元组$$ δ(-\infty), δ(0)=z_0, δ(1)=z_1 , δ(+\infty), $$的交比之间的关系，其中$δ$是${\cal D}$的唯一测地线，分别在时间$t=0$和$t=1$连接$z_0$和$z_1$。
英文摘要：
We study the Poincaré 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 $$ δ(-\infty), δ(0)=z_0, δ(1)=z_1 , δ(+\infty), $$ where $δ$ is the unique geodesic of ${\cal D}$ joining $z_0$ and $z_1$ at times $t=0$ and $t=1$, respectively.
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