Jordan-morphisms in $\ast$-algebras

Abstract:

As a continuation of Størmer’s work on Jordan-morphisms in $C*$-algebras we consider Jordan-morphisms $\varphi$ from $*$-algebras $\mathfrak {A}$ into the $*$-algebra $B(\mathcal {H})$, and assume that $\varphi (\mathfrak {A})$ is again a $*$-algebra. We then establish the existence of three mutually orthogonal central projections ${P_i}$, $i = 1,2,3$, in $\varphi {\left ( {} \right )^{\prime \prime }}$ such that ${P_1} + {P_2} + {P_3} = I$ and $\varphi ( \cdot ){P_1}$ is a morphism, $\varphi ( \cdot ){P_2}$ is an antimorphism. ${P_3}$ is the largest projection such that $\varphi ( \cdot ){P_3}$ 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.