Spectrum preserving linear mappings for scattered Jordan-Banach algebras

Abstract

Given two semisimple complex Jordan-Banach algebras with identity A A and B B , we say that T T is a spectrum preserving linear mapping from A A to B B if T T is surjective and we have Sp ⁡ ( T x ) = Sp ⁡ ( x ) \operatorname {Sp}(Tx)=\operatorname {Sp}(x) , for all x ∈ A x\in A . We prove that if B B is a scattered Jordan-Banach algebra, then T T is a Jordan isomorphism.