Projective completions of Jordan pairs. Part II: Manifold structures and symmetric spaces

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ``compact-like'' duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as ``standard models'' - they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra.

(This paper has appeared in Geometriae Dedicata 112 (2005), 73--113)

Wolfgang Bertram < bertram@iecn.u-nancy.fr >

Karl-Hermann Neeb < neeb@mathematik.tu-darmstadt.de >