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.