A non commutative K\"ahler structure on the Poincar\'e disk of a C*-algebra

We study the Poincar\'e disk $\d=\{z\in\a: \|z\|<1\}$ of a C$^*$-algebra $\a$ as a homogeneous space under the action of an appropriate Banach-Lie group $\u(\theta)$ of $2\times 2$ matrices with entries in $\a$. We define on $\d$ a homogeneous K\"ahler structure in a non commutative sense. In particular, this K\"ahler structure defines on $\d$ a homogeneous symplectic structure under the action of $\u(\theta)$. This action has a moment map that we explicitly compute. In the presence of a trace in $\a$, we show that the moment map has a convex image when restricted to appropriate subgroups of $\u(\theta)$, resembling the classical result of Atiyah-Guillmien-Sternberg.