On the Fischer-Muszély equation for the positive cones of $C^{\ast }$-algebras

We study the Fischer-Muszély functional equation for the positive semidefinite and the positive definite cones of unital $C^{\ast }$-algebras. We show that any bijection between the positive semidefinite cones satisfying the Fischer-Muszély equality extends to a Jordan $\ast$-isomorphism followed by multiplication on both sides by a positive element. As a corollary, we obtain a similar result for the positive definite cones of unital $C^{\ast }$-algebras.