The Poincaré Half-Space of a C*-Algebra

Abstract

In this chapter we continue our study of the space H and its alternative presentations as Qρ and D. Regarded as D, the projective nature of 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 δ(−∞), δ(0) = z0, δ(1) = z1, δ(+∞) induced by a geodesic δ of D, coincides with the logarithm based at z_0 z0 evaluated at the point z1 (as happens in the classic scalar Poincaré disk). We introduce in H an operator valued Kähler structure. This, in turn, induces on 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 H. We compute the moment map. In the presence of a trace in A, we show that, for an appropriate subgroup of U(θ), the image of the moment map is a convex set, which resembles the classical result of Atiyah, Guillemin and Sternberg.