  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.
