Additivity of Jordan$^ \ast$-maps on $AW^ \ast$-algebras

Abstract:

Let $M$ and $N$ be AW*-algebras and $\phi$ be a Jordan*-map from $M$ to $N$ which satisfies (1) $\phi (x \circ y) = \phi (x) \circ \phi (y)$ for all $x$ and $y$ in $M$, (2) $\phi ({x^*}) = \phi {(x)^*}$ for all $x \in M$, and (3) $\phi$ is bijective, where $x \circ y = (1/2)(xy + yx)$. If $M$ has no abelian direct summand and a Jordan*-map $\phi$ is uniformly continuous on every abelian ${C^*}$-subalgebra of $M$, then we can conclude that $\phi$ is additive. Moreover, $\phi$ is the sum of ${\phi _i}(i = 1,2,3,4)$ such that ${\phi _1}$ is a linear $*$-ring isomorphism, ${\phi _2}$ is a linear $*$-ring anti-isomorphism, ${\phi _3}$ is a conjugate linear $*$-ring anti-isomorphism and ${\phi _4}$ is a conjugate linear $*$-ring isomorphism.