In this chapter we continue our study of the space \(\mathcal {H}\) and its alternative presentations as \(\mathcal {Q}_\rho \) and \(\mathcal {D}\) . Regarded as \(\mathcal {D}\) , the projective nature of \(\mathcal {H}\) arises: it is using this model that we can characterize the limit points of geodesics, and prove one of our main results: the (operator valued) cross-ratio of the four-tuple \(\delta (-\infty ), \delta (0)=z_0, \delta (1)=z_1, \delta (+\infty )\) induced by a geodesic \(\delta \) of \(\mathcal {D}\) , coincides with the logarithm based at \(z_0\) evaluated at the point \(z_1\) (as happens in the classic scalar Poincaré disk). We introduce in \(\mathcal {H}\) an operator valued Kähler structure. This, in turn, induces on \(\mathcal {H}\) a homogeneous symplectic structure, which is the imaginary part of the Hilbertian form. We show that this symplectic structure is the covariant differential of the Liouville 1-form of \(\mathcal {H}\) . We compute the moment map. In the presence of a trace in \(\mathcal {A}\) , we show that, for an appropriate subgroup of \(\mathcal {U}(\theta )\) , the image of the moment map is a convex set, which resembles the classical result of Atiyah, Guillemin and Sternberg.