Symmetric Cones And Euclidean Jordan Algebras
Jacques Faraut, Adam Korányi · 1994
Abstract In This Chapter We Introduce Euclidean Jordan Algebras. As Will Be Seen In Chapter Viii, These are the same as the formally real Jordan algebras more usually considered in the literature. For such algebras we prove a ‘spectral theorem’, which, in the case of the algebra of real symmetric matrices, specializes to the usual spectral theorem. Then we prove a fundamental result which establishes a one-to-one correspondence between symmetric cones and Euclidean Jordan algebras. A consequence of this is the unique decomposition of every symmetric cone into a direct product of irreducible ones. Finally, for symmetric cones, we give an independent description in Jordan algebra terms of all the geometric and analytic objects introduced in Chapter I.