Involution-preserving ring isomorphisms in norm between unital $C^*$-algebras

Abstract

Let $A$ and $B$ be nonzero unital $C^*$-algebras with units $1_A$ and $1_B$, respectively, and let $T\colon A\to B$ be a bijection satisfying \[ \|T(a+b)\|=\|T(a)+T(b)\|, \qquad \|T(ab)\|=\|T(a)T(b)\|, \qquad T(a^*)=T(a)^* \] for all $a,b\in A$. We prove that there exist a central symmetry $u$ in $B$ and a real $*$-isomorphism $Φ\colon A\to B$ such that \[ T(a)=uΦ(a) \qquad(a\in A). \] Moreover, $u$ and $Φ$ are uniquely determined by $T$. Conversely, if $u\in B$ is a central symmetry and $Φ\colon A\to B$ is a real $ * $-isomorphism, then $T=uΦ$ is a bijection satisfying the three identities above. In particular, the two norm identities, together with involution preservation, force the normalized map $uT$ to preserve the full product, not merely the Jordan product.