On Mappings That Preserve the Jordan Product: A Revisited Study

We study bijective maps between algebras that preserve Jordan-type products. More precisely, given algebras A and B, we consider maps Θ:A→B satisfying Θ(AB+BA)=Θ(A)Θ(B)+Θ(B)Θ(A) or, more generally, preserving the Jordan triple product Θ({A,B,C})={Θ(A),Θ(B),Θ(C)}, where {A,B,C}=ABC+CBA. We first show that if A is a prime unital algebra with a nontrivial idempotent and Θ is a bijection of the above form, then Θ is necessarily additive. This extends known additivity results for Jordan maps and Jordan triple maps on standard operator algebras and matrix algebras. We then prove that if, in addition, both A and B are prime algebras, any bijective map preserving the Jordan triple product is either multiplicative or anti-multiplicative. Equivalently, such a map is an isomorphism or an anti-isomorphism. These results unify and generalize earlier work on Jordan maps, Jordan triple maps, and structure-preserving transformations on prime rings and standard operator algebras.