Surjective Fischer--Musz\'ely maps on positive cones of JB-algebras

Abstract

We prove that every surjective Fischer--Musz\'ely map between positive cones of arbitrary JB-algebras is additive and positively homogeneous, and extends uniquely to a bounded positive linear surjection. In particular, this gives a complete affirmative answer to Moln\'ar's problem for arbitrary $C^*$-algebras. The theorem includes nonunital and exceptional JB-algebras and assumes neither injectivity nor continuity. The proof connects the norm-valued functional equation to positive-cone rigidity through translation invariance of an induced pseudometric. For a surjective map satisfying the Fischer--Musz\'ely identity with uniform error $\varepsilon$, we also construct a unique bounded positive linear map $L$ at uniform distance at most $3\varepsilon$. The set $L(A_+)$ is dense in $B_+$, although $L$ need not be surjective, even when the original map is continuous and bijective. For bijective FM maps, we obtain weighted Jordan representations on the biduals. We also extend the characterization of surjective norm-sum preservers to JB-algebras and characterize norm arithmetic-midpoint and harmonic-mean identities on positive invertible cones of unital JB-algebras.