TRACE AND SPECTRUM PRESERVING LINEAR MAPPINGS IN JORDAN-BANACH ALGEBRAS

In the first section we define the trace on the socle of a Jordan-Bana ch algebra in a purely spectral way and we prove that it satisfies sev eral identities. In particular this trace defines the Faulkner bilinea r form. In the second section, using analytic tools and the properties of the trace, we prove that a spectrum preserving linear mapping from J onto J', where J and J' are semisimple Jordan-Banach algebras, is n ot far from being a Jordan isomorphism. It is in particular a Jordan i somorphism if J' is primitive with non-zero socle.