These notes were written following lectures I had the pleasure of giving on
this subject at Keio University, during November and December 2004. The first
part is about new applications of Jordan algebras to the geometry of Hermitian
symmetric spaces and to causal semi-simple symmetric spaces of Cayley type. The
second part will present new contributions for studing (non commutative) Hardy
spaces of holomorphic functions on Lie semi-groups which is a part of the so
called Gelfand-Gindikin program
