Poincaré's disk and half-space of a C*-algebra as homogeneous spaces

It is shown that both the Poincaré disk and the Poincaré half-space of a C*-algebra A have a structure of reductive homogeneous spaces and as such they can be immersed in the space of positive operators of 2-by-2 matrices on A, which have non-curvature positive properties in the sense of Alexandroff.