  Cirelli, Mani\`{a} and Pizzocchero generalized quantum mechanics by
K\"{a}hler geometry. Furthermore they proved that any unital C$^{*}$-algebra is
represented as a function algebra on the set of pure states with a
noncommutative $*$-product as an application. The ordinary quantum mechanics is
regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal
H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an
infinite dimensional K\"{a}hler manifold of positive constant holomorphic
sectional curvature. In general, such dynamical system is constructed for a
general K\"{a}hler manifold of nonzero constant holomorphic sectional curvature
$c$. The Hilbert ball $B_{{\cal H}}$ is defined by the open unit ball in ${\cal
H}$ and it is a K\"{a}hler manifold with $c<0$. We introduce the quantum
mechanics on $B_{{\cal H}}$. As an application, we show the structure of the
noncommutative function algebra on $B_{{\cal H}}$.
