On non-commutative Minkowski spheres

The purpose of the following is to try to make sense of the stereographic projection in a non-commutative setup. To this end, we consider the open unit ball of a ternary ring of operators, which naturally comes equipped with a non-commutative version of a hyperbolic metric and ask for a manifold onto which the open unit ball can be mapped so that one might think of this situation as providing a noncommutative analog to mapping the open disk of complex numbers onto the hyperboloid in three space, equipped with the restriction of the Minkowskian metric. We also obtain a related result on the Jordan algebra of self-adjoint operators.