As a continuation of Størmer’s work on Jordan-morphisms in C ∗ C* -algebras we consider Jordan-morphisms φ \varphi from ∗ * -algebras A \mathfrak {A} into the ∗ * -algebra B ( H ) B(\mathcal {H}) , and assume that φ ( A ) \varphi (\mathfrak {A}) is again a ∗ * -algebra. We then establish the existence of three mutually orthogonal central projections P i {P_i} , i = 1 , 2 , 3 i = 1,2,3 , in φ ( ) ′ ′ \varphi {\left ( {} \right )^{\prime \prime }} such that P 1 + P 2 + P 3 = I {P_1} + {P_2} + {P_3} = I and φ ( ⋅ ) P 1 \varphi ( \cdot ){P_1} is a morphism, φ ( ⋅ ) P 2 \varphi ( \cdot ){P_2} is an antimorphism. P 3 {P_3} is the largest projection such that φ ( ⋅ ) P 3 \varphi ( \cdot ){P_3} is a morphism, as well as an antimorphism. Uniqueness is also shown. The theorem improves a result of Kadison and Størmer. Our proofs are self-contained.